Defusing Bertrand’s Paradox

Zalán Gyenis
Department of Mathematics and its Applications

Central European University
Nádor u. 9. H-1051 Budapest, Hungary

gyz@renyi.hu

Miklós Rédei ∗

Department of Philosophy, Logic and Scientific Method
London School of Economics and Political Science

Houghton Street, London WC2A 2AE, UK
m.redei@lse.ac.uk

Forthcoming in
The British Journal for the Philosophy of Science

Abstract

The classical interpretation of probability together with the Principle of Indifference are formu-
lated in terms of probability measure spaces in which the probability is given by the Haar measure. A
notion called Labeling Invariance is defined in the category of Haar probability spaces, it is shown that
Labeling Invariance is violated and Bertrand’s Paradox is interpreted as the very proof of violation
of Labeling Invariance. It is shown that Bangu’s attempt [2] to block the emergence of Bertrand’s
Paradox by requiring the re-labeling of random events to preserve randomness cannot succeed non-
trivially. A non-trivial strategy to preserve Labeling Invariance is identified and it is argued that,
under the interpretation of Bertrand’s Paradox suggested in the paper, the paradox does not under-
mine either the Principle of Indifference or the classical interpretation and is in complete harmony
with how mathematical probability theory is used in the sciences to model phenomena; it is shown
in particular that violation of Labeling Invariance does not entail that labeling of random events
affects the probabilities of random events. It also is argued however that the content of the Principle
of Indifference cannot be specified in such a way that it can establish the classical interpretation of
probability as descriptively accurate or predictively successful.

1 The main claims

Bertrand’s Paradox, published first in [3], is regarded a classical problem in connection with the
classical interpretation of probability based on the Principle of Indifference, and it continues to
attract interest [17], [24], [2], [22] in spite of alleged resolutions that have been suggested in the
large and still growing literature discussing the issue ([11] and [16] are perhaps the most well-known
suggestions for resolutions; the Appendix in [16] contains a brief summary of a number of typical
views of the Paradox).

It is not the aim of this paper to offer yet another “resolution” or criticize the ones available;
rather, we suggest a new interpretation of Bertrand’s Paradox and analyze its relation to the classical
interpretation of probability. The interpretation proposed here should make clear that Bertrand’s
Paradox cannot be “resolved” – not because it is an unresolvable, genuine paradox but because there
is nothing to be resolved: the “paradox” simply states a provable, non-trivial mathematical fact, a
fact which is perfectly in line both with the correct intuition about how probability theory should be
used to model phenomena and with how probability theory is applied in the sciences.

∗Acknowledgement: research supported in part by the Hungarian Scientific Research Found (OTKA). Contract number:
K68043.

1



The key idea of the interpretation to be developed here is that the category of probability mea-
sure spaces with an infinite set of random events for which a classical interpretation of probability
based on the Principle of Indifference can be meaningfully formulated is the one in which the
set X of elementary events is a compact topological group, the Boolean algebra S representing the
set of random events is the set of Borel subsets of X and the probability measure pH is the (nor-
malized) Haar measure on S. After stating the General Classical Interpretation in terms of the
probability measure space (X,S,pH) together with the Principle of Indifference, we will define
a notion called Labeling Invariance in this category of measure spaces: Labeling Invariance
states that a re-labeling (X′,S′) of random events (X,S) is an isomorphism between (X,S,pH) and
(X′,S′,p′H). Labeling Invariance can be interpreted as the expression (in the context of the clas-
sical interpretation) of the general intuition we call Labeling Irrelevance: that the specific way the
random events are named is irrelevant from the perspective of the value of their probability. It will be
shown that Labeling Invariance does not hold in the category of Haar probability measure spaces
and we interpret Bertrand’s Paradox as stating this provable mathematical fact. We will also argue
however that Labeling Invariance is not the proper way to express Labeling Irrelevance: our
freedom to choose measure theoretically isomorphic probability theories to describe the same random
phenomenon manifests the conventionality of naming random events in probabilistic modeling; thus
violation of Labeling Invariance is perfectly compatible with Labeling Irrelevance.

This interpretation of Bertrand’s Paradox makes it possible to formulate precisely the extra condi-
tion on re-labelings that ensures that re-labelings do preserve the probabilities of events; the condition
is an expression of the demand that re-labelings do not affect our epistemic status about the elemen-
tary events. We also will show that the recent attempt by Bangu [2] to block the emergence of
Bertrand’s Paradox by requiring re-labelings to preserve randomness cannot succeed non-trivially.

The interpretation will also make it clear that Bertrand’s Paradox does not affect the Principle of
Indifference and does not, in and by itself, undermine the classical interpretation of probability – the
classical interpretation, the Principle of Indifference and Labeling Invariance are independent
ideas. This is not to say that the classical interpretation is maintainable however; the main problem
with it is that it gives the impression that it is possible to infer empirically correct probabilities from
an abstract principle stating some sort of epistemic neutrality. It would be a mystery if this were
possible, but we will argue in the final section that this is not possible and does not in fact happen
in applications of probability theory.

2 The elementary classical interpretation of probability

Betrand’s Paradox appeared at a time when probability theory had already progressed from the
purely combinatorial phase involving only a finite number of random events to the period when it got
intertwined with calculus. This development began in the early 18th century with the appearance of
limit theorems (theorem of large numbers, Bernoulli 1713, and central limit theorem, de Moivre 1733,
[7]); yet, by the late 19th century the theory had not yet reached the maturity that would have made
the mathematical foundations of the theory clear and transparent. This was clearly recognized by
Hilbert, who, in his famous lecture in Paris in 1900, mentioned the need of establishing probability
theory axiomatically as one of the important open problems (Hilbert’s 6th problem [28], [27][p.
32-36]). Hilbert’s call was answered only in 1933, when Kolmogorov firmly anchored probability
theory within measure theory [13]. (See [6] for the history of some of the major steps leading to the
Kolmogorovian axioms.)

In the measure theoretic approach probability theory is a triplet (X,S,p), where X is the set of
elementary random events, S (the set of general random events) is a Boolean σ algebra of certain
subsets of X and p (the probability) is a countably additive measure from S into the unit interval
[0, 1]. Typically, one also needs random variables to describe certain features of the phenomenon to
be described probabilistically: A (real valued) random variable f is a measurable function f from X
into the set of real numbers R; measurability being the requirement that the inverse image f−1(d)
of any Borel set d in R belongs to S. The measurability requirement entails that the distribution
of a random variable d 7→ p(f−1(d)) is well-defined, the distribution of f is in fact the probability
measure p◦f−1 on B(R) defined as (p◦f−1)(d) = p(f−1(d)) for all Borel sets d ∈B(R). The number
p(f−1(d)) is the probability that f takes its value in d. Note that the events also can be regarded as
random variables: an element A in S can be identified with the characteristic (also called: indicator)
function χA of the set A (see e.g. [21] for the mathematical notions of measure theoretic probability).

Remark 1. One should note that the Kolmogorovian axioms are not the only axiomatization of prob-
ability theory: Rényi axiomatized probability theory by taking the concept of conditional probability
as primitive [20]; whereas conditional probability is a defined concept in Kolmogorovian axiomatiza-
tion. Popper’s axiomatization [19] takes relative probability as primitive (see [15] for a recent analysis

2



of Renyi’s and Popper’s concept). Keynes’ replacing the interval [0, 1] as range of probability by a
more general partially ordered set [12] also can be considered as an axiomatization different from the
Kolmogorovian one. These axiomatizations notwithstanding, the Kolmogorovian axiomatization is
by far the most important and it has been widely accepted as the mathematical theory of probability:
All the major mathematical results on probability theory are obtained in this framework and the suc-
cessful probabilistic models in the sciences also are created in terms of measure theoretic probability
theory (for instance: classical statistical mechanics, theory of stochastic processes). One also should
note the following:

• Probability theory understood as the triplet (X,S,p) specified by the Kolmogorovian axioms
is part of pure mathematics; hence “measure theoretic probability” (understood as (X,S,p)) is
not an interpretation of probability and must thus not be contrasted with any interpretation of
probability. More will be said on the relation of (X,S,p) to interpretations in section 7.

• “Measure theoretic probability” (the triplet (X,S,p)) also describes mathematically the elemen-
tary situations where the set of random events is finite. These include the classical, paradigm
examples of random phenomena (all sorts of gambling such as coin flipping, die throwing, card
games). To describe these was the main motivation to develop probability theory in the first
place. On mild additional conditions, measure theoretic probability also describes the betting
situations in which the events to bet on are identifiable with propositions stating that the events
happened (or not) – the mild condition is that the propositions are closed under the elementary
logical operations, i.e. that they form a Boolean algebra.

The significance of probability theory being part of measure theory is that foundational-conceptual
problems of probability theory, such as Bertrand’s Paradox, can best be analyzed in terms of measure
theoretic concepts. With few exceptions, the papers on Bertrand’s Paradox typically do not aim at
providing an analysis on this level of abstraction however, and, as a result, the precise nature of the
paradox remains less clear than it should be. One such exception is Shackel’s paper [24], which raises
the issue of “Getting the level of abstraction right” [24][p. 156] explicitly. But the level of abstraction
suggested by Shackel is a bit too high. To see why, we recall first the classical interpretation of
probability together with the Principle of Indifference in measure theoretic terms.

The elementary version of the classical interpretation of probability concerns the probability space
(Xn,P(Xn),pu), where Xn is a finite set containing n number of random events and the full power
set P(Xn) of Xn represents the set of all events. The probability measure pu is determined by the
requirement that the probability pu(A) be equal to the ratio of the “number of favorable cases to the
number of all cases”:

pu(A) =
number of elements in the set {xi : xi ∈ A}

n
(1)

This is equivalent to saying that pu is the probability measure that is uniform on the set of elementary
events.

While it is not always stated and emphasized explicitly, it also is part of the classical interpretation
what we call here the Interpretive Link: that the numbers pu(A) are related to something non-
mathematical. Without such an interpretive link, the classical interpretation is not an interpretation
of probability at all: the numbers pu(A) defined by (1) are just pure, simple mathematical relations.
There are two standard Interpretive Links: The Frequency Link and the Degree of Belief
Link. We restrict the discussion to the Frequency Link because the classical interpretation emerged
historically and was formulated on the basis of this link:

Elementary Classical Interpretation: In case of a finite number of elementary events the prob-
abilities of events are given by the measure pu that is uniform on the set of elementary events, and
(Frequency Link:) the numbers pu(A) will be (approximately) equal to the relative frequency of A
occurring in a series of trials producing elementary random events from Xn.

Notice the future tense in the above formulation: it is this reference for future random trials that
distinguishes the classical interpretation (with the Frequency Link) from the frequency interpretation,
in which the ensemble of elementary random events determining A’s relative frequency must be
specified before one can talk about probabilities (cf. [26][p. 24]).

The classical interpretation so formulated is not maintainable however: simple examples (such as
throwing a loaded die) show that it is only under special circumstances that pu(A) is indicative of
the frequencies with which A will occur in trials. This is what the Principle of Indifference is
supposed to express. To state this principle we reformulate first the condition (1). Let Πn be the
group of permutations of the n element set {1, 2, . . .n} and π ∈ Πn be a permutation. Then the
probability measure pu on P(Xn) which is uniform on Xn is determined uniquely by the condition

for every π ∈ Πn one has: pu({xi}) = pu({xπ(i)}) for all i ∈{1, 2, . . .n} (2)

3



Elementary Principle of Indifference: If the permutation group Πn expresses epistemic indiffer-
ence about the elementary random events in Xn, then the (Elementary) Classical Interpretation
is correct.

Thus the (Elementary) Principle of Indifference states that the (elementary version of the)
classical interpretation of probability is maintainable only if one is epistemically neutral in some
sense about the elementary events. For now, we leave open the question of how the content of the
“epistemic neutrality” should be specified in order for the Principle of Indifference to hold; we
will return to the issue of epistemic neutrality in section 7.

3 The general classical interpretation of probability in
terms of Haar measures

Bertrand’s Paradox is typically regarded as an argument against the universal applicability of the
Principle of Indifference: Bertrand’s Paradox type arguments are intended to show that applying
the Principle of Indifference can lead to assigning different probabilities to the same event. Both
the original version of the argument and the numerous simplified versions of it involve an (uncount-
ably) infinite number of elementary random events however. But then it is not obvious at all how
one can apply the Principle of Indifference because the formulation of it in the previous section
looses its meaning if the set of elementary events is not finite: there is no permutation group in the
infinite case with respect to which one could require invariance of the measure yielding the “right”
probabilities; equivalently: there is no probability measure on an infinite S that would be uniform
on the infinite set X of elementary events. What is then the Principle of Indifference in connec-
tion with such infinite probability spaces? Without answering this question in suitable generality,
Bertrand’s Paradox cannot be properly discussed in measure theoretic concepts.

Shackel’s paper [24], which aims at an analysis of Bertrand’s Paradox in abstract measure the-
oretic terms, realizes the importance of this question but does not offer a convincing specification
of the Principle of Indifference: Shackel just assumes a measure µ on S and stipulates that the
probabilities p(A) be given by µ as p(A) = µ(A)/µ(X) (“Principle of indifference for continuum sized
sets” [24][p. 159]). But there are infinitely many measures µ on S that could in principle be taken
as ones that define a probability p. Which one should be singled out that yields a p that could
in principle be interpreted as expressing epistemic indifference about elements in X? This crucial
question remains unanswered in [24].

It is clear that without some further structure on an infinite X it is not possible to single out any
probability measure on S and hence it is impossible to formulate an indifference principle on such a
measurable space. The formulation of the Elementary Principle of Indifference in terms of the
permutation group Πn gives a hint about what kind of structure is needed in the more general case
however: It is a natural idea to try to replace the permutation group Πn by another group G to be
interpreted as expressing epistemic neutrality and hope that the elements g of G determine a function
αg : X → X (an action on X) in such a way that if one requires the analogue of (2) by postulating

for all g ∈G : p∗(A) = p∗(αg[A]) for all A ∈S (3)

then the above condition (3) determines a unique probability measure p∗ on S, just like in the case
of a finite number of events. Problem is that for a general measurable space (X,S) with a continuum
sized X there is no guarantee in general that a G exist leading to a p∗ – much less that it leads to
a unique p∗. There is however such a guarantee under some additional assumptions: If X itself is a
topological group satisfying certain conditions.

If X is a locally compact abelian topological group, or a not necessarily abelian but compact
topological group, then there exists a unique (up to multiplication by a constant) measure (called:
the Haar measure) pH on (the Borel sets of) X which is invariant with respect to the group action.
Furthermore, if X is compact then the measure pH is normalized and pH is then a probability
measure. (The Appendix collects some elementary facts about the Haar measure; equation (29) in
the Appendix formulates the invariance of the Haar measure precisely).

The canonical example of an unbounded Haar measure is the Lebesgue measure on the real line:
the Lebesgue measure is the unique measure on the real line that is invariant with respect to the
real numbers as an additive group – the group action is the shift on the real line. The same holds
for the Lebesgue measure on Rn. The normalized restrictions of the Lebesgue measure on Rn to
bounded, compact subsets of Rn are thus distinguished by the feature that they originate from a
shift-invariant measure; moreover, the Lebesgue measure on any interval [a,b] also can be regarded as
Haar measure in its own right and the same holds for sets ×ni [ai,bi] in Rn (cf. Appendix). Both the
original Bertrand’s Paradox and all of the simplified versions of it take the normalized restriction of

4



the Lebesgue measure to some bounded, compact sets in Rn (n = 1, 2) as the measure that expresses
the Principle of Indifference. This amounts to interpreting (more or less tacitly) the group that
generates the Lebesgue measure as a symmetry expressing epistemic neutrality about the elementary
random events.

Thus, in general, the group action on X determined by X itself as a group can play the role of the
action of the permutation group on Xn, and the Haar measure pH on a compact X is the analogue
of the uniform distribution on Xn if a non-zero uniform distribution on the elements X does not
exist, which is the case if X is an infinite set. Note that taking the Haar measure as the analogue
of the uniform distribution is also justifiable using maximum entropy techniques (see [10]). In what
follows, (X,S,pH) stands for a probability measure space in which X is a compact topological group
with continuous group action, S is the Borel σ algebra on X and pH is the Haar measure on S.
In the terminology of these group and measure theoretic notions the general classical interpretation
of probability and the related principle of indifference can be consistently formulated generally as
follows:

General Classical Interpretation: If X is a compact topological group, then the probabilities of
the events are given by the Haar measure pH on (the Borel sets of) X, and (Frequency Link:) the
numbers pu(A) will be (approximately) equal to the relative frequency of A occurring in a series of
trials producing elementary random events from X.

General Principle of Indifference: If X is a compact topological group and if the group action
expresses epistemological indifference about the elementary random events in X, then the General
Classical Interpretation is correct.

4 Labeling Invariance and Labeling Irrelevance

Probabilities understood in the spirit of the elementary classical interpretation based on the Princi-
ple of Indifference have the property we call here Labeling Invariance: the probability measure
which is permutation invariant in one labeling assigns the same value of probability to the random
events as the probability measure that is permutation invariant in a different labeling of the same
random events. For instance, if one has a symmetric die sides of which are numbered by numbers
1, 2, . . . 6, then the permutation invariant measure on the elementary events is the uniform probabil-
ity measure that assigns the value 1

6
to each side. If, instead of labeling the sides by the numbers

1, 2, . . . 6, we label them by painting them using the colours red, blue, green, yellow, black, white,
say, then the measure which is invariant with respect to the permutation of colours will assign the
value 1

6
to each colour, i.e. the same value that was assigned to the sides on the basis of the labeling

using the numbers 1, 2, . . . 6 as labels.
Labeling Invariance seems to be an important and attractive feature of probabilities interpreted

according to the elementary classical interpretation because it entails that Labeling Irrelevance
holds for the elementary classical interpretation: Labeling Irrelevance states that labeling of ran-
dom events does not affect their probabilities, that a particular labeling of random events is a matter
of convention. Labeling Irrelevance is an important condition in probabilistic modeling: its viola-
tion would entail a radical ambiguity and arbitrariness in assigning probabilities to random events.
Without Labeling Irrelevance one could not collect statistical data, and violation of Labeling
Irrelevance is obviously incompatible with any interpretation of probability that treats probability
as an objective feature of the world; furthermore, violation of Labeling Irrelevance also would
make subjective degrees of belief vulnerable to a Dutch book: if degrees of belief would change as a
result of re-naming the random events involved, then the bookie could make money by just re-naming
the events in the bet. Given this conceptual importance of Labeling Irrelevance and the fact that
Labeling Invariance entails it, one expects Labeling Invariance to hold for the general classical
interpretation as well. We well see shortly that Bertrand’s Paradox can be viewed as proof that
Labeling Invariance does not hold in the general classical interpretation. It should be emphasized
however that this does not entail that Labeling Irrelevance cannot be maintained in general, nor
does it follow from violation of Labeling Invariance that the classical interpretation based on the
Principle of Indifference is inconsistent because Labeling Invariance and Labeling Irrele-
vance are not equivalent. We will say more on the relation of Labeling Invariance, Labeling
Irrelevance and probabilistic modeling in section 7.

To formulate the idea of Labeling Invariance generally and precisely, we need the notion of
re-labeling (re-naming) first: If (X,S,pH) and (X′,S′,p′H) are two probability spaces describing the
same phenomenon, then the map h: X → X′ is called a re-labeling if it is a bijection between X and

5



X′ and both h and its inverse h−1 are measurable, i.e. it holds that

h[A] ∈S′ for all A ∈S (4)
h
−1

[B] ∈S for all B ∈S′ (5)

(Here h[A] = {h(x) : x ∈ A} and h−1[A′] = {h−1(x′) : x′ ∈ A′}.) Note that without the measurability
condition required of h it can happen that a general event A ∈ S has probability but its re-named
version h[A] does not – in this case h cannot be called re-naming of random events (and similarly for
h′).

Labeling Invariance is the claim that the probabilities understood in the spirit of the classical
interpretation are invariant with respect to re-naming; that is to say, if (X,S,pH) and (X′,S′,p′H)
are two probability spaces and h is a re-labeling between X and X′ then it holds that

p
′
H(h[A]) = pH(A) for all A ∈S (6)

pH(h
−1

[A
′
]) = p

′
H(A

′
) for all A

′ ∈S′ (7)

Recall (see e.g. [1][p. 3]) that two probability measure spaces (X,S,p) and (X′,S′,p′) are called
isomorphic if there are sets Y ∈ S and Y ′ ∈ S′ such that p(Y ) = 0 = p′(Y ′) and there exists a
bijection f : (X \Y ) → (X′ \Y ′) such that both f and its inverse f−1 are measurable and such that
both f and f−1 preserve the measure p and p′, respectively; i.e. (8)-(9) below hold:

p
′
(f[A]) = p(A) for all A ∈S (8)

p(f
−1

[A
′
]) = p

′
(A
′
) for all A

′ ∈S′ (9)

The function f is called then an isomorphism between the probability measure spaces. Labeling
Invariance can therefore be expressed compactly by saying

Labeling Invariance: Any re-labeling between probability spaces (X,S,pH) and (X′,S′,p′H) is an
isomorphism between these probability spaces.

5 General Bertrand’s Paradox

Labeling Invariance is obviously a very strong claim and Bertrand’s paradox can be interpreted
as the proof that it cannot be maintained in general (see below). But why would one think that
Labeling Invariance holds in the first place? The answer is: because Labeling Invariance does
hold for an infinite number of probability spaces: for probability spaces with any finite number
elementary random events probabilities of which are given by the uniform probability measure. A
bijection h between two finite sets Xn and X

′ = Xm of elementary events exists if and only if the
sets Xn and Xm have the same number of elements, n = m, and this entails that the two uniform
distributions on those equivalent sets will assign the same probability to A and h[A] (and to A′

and h−1[A′]) – no Bertrand’s Paradox can arise in this case. Since the intuition about probability
theory was shaped historically by situations involving only a finite number of random events, it is
not surprising that Labeling Invariance became part of the intuition about probability. It turns
out however that this intuition is a poor guide if the set of elementary events is not finite: This is
precisely what Bertrand’s Paradox shows, general form of which is the following statement:

General Bertrand Paradox: Let (X,S,pH) and (X′,S′,p′H) be probability spaces with compact
topological groups X and X′ having an infinite number of elements and pH,p

′
H being the respective

Haar measures on the Borel σ algebras S and S′ of X and X′. Then Labeling Invariance does not
hold for (X,S,pH) and (X′,S′,p′H) in the sense that
• either there is no re-labeing between X and X′;
• or, if there is a re-labeling between X and X′, then there also exists a re-labeling that violates

Labeling Invariance.

The General Bertrand’s Paradox is a trivial consequence of the following non-trivial theorem in
measure theory:

Proposition 1 ([25], [23]). If X is an infinite, compact topological group with the Haar measure pH
on the Borel σ algebra S of X, then there exists an autohomeomorphism θ of X and an open set E
in S such that pH(θ[E]) 6= pH(E).

6



By definition an autohomeomorphism θ of X is a bijection from X into X such that both θ and its
inverse θ−1 are continuous. Since continuous functions are Borel measurable, an autohomeomorphism
is a re-labeling: a re-labeling of X in terms of its own elements. Assume now that (X,S,pH) and
(X′,S′,p′H) are two probability spaces with infinite, compact topological groups X and X

′ and Haar
measures pH and p

′
H. If h: X → X

′ is a re-labeling between X and X′ then either h is an isomorphism
between the probability spaces (i.e. preserves the probability in the sense of (6)-(7)) or it is not. If
it is not, then Labeling Invariance is violated by h. If h does preserve the probability (and is
thus an isomorphism between (X,S,pH) and (X′,S′,p′H)) then by Proposition 1 there exists an
autohomeomorphism θ on X and there exists an open set E ∈S such that pH(θ[E]) 6= pH(E). This
means that for the re-labeling given by the composition h◦θ we have

p
′
H((h◦θ)[E]) = p

′
H(h[θ[E]]) = pH(θ[E]) 6= pH(E) (10)

so the re-labeling h ◦ θ violates (6) and thus h ◦ θ violates Labeling Invariance. In either case
Labeling Invariance is violated. Furthermore, the autohomeomorphism ensured by Proposition 1
provides a re-labeling of the elementary set of events of any infinite compact group in terms of its
own elementary events in such a way that the Haar measure yielding the probabilities of the events
in the spirit of the classical interpretation are not preserved under the re-labeling.

The General Bertrand’s Paradox is thus a general feature of infinite probability measure spaces
with the Haar measure yielding the probabilities, and note that it says more than the original
Bertrand’s Paradox, which only claimed that there exist Haar measures and re-labelings that violate
Labeling Invariance: The General Bertrand’s Paradox says that no two infinite Haar probability
spaces can satisfy Labeling Invariance; i.e. if there is at all a re-labeling between two probability
spaces (X,S,pH) and (X′,S′,p′H) with infinite X and X

′ then there is also a re-labeling between
these spaces that violates Labeling Invariance, and for any space (X,S,pH) with an infinite X
there exists a space (namely itself) and a self-re-labeling of (X,S,pH) that violates Labeling In-
variance. Thus Bertrand’s 1888 Paradox can be viewed as the specific “Lebesgue measure case” of
a mathematical theorem that was proved in full generality in 1993 only.

We close this section by giving an explicit, elementary example of violation of Labeling Invari-
ance; this example will be referred to in the next section. In a well-defined sense (explained in
Remark 2) the example is general.

Example Let [a,b] and [c,d] be two closed intervals of the real numbers and

([a,b],S[a,b],p[a,b]) and ([c,d],S[c,d],p[c,d])
be the two probability spaces with p[a,b] and p[c,d] being the normalized Lebesgue measures on the
intervals [a,b] and [c,d], with S[a,b] and S[c,d] being the Borel measurable sets of the respective
intervals. Elementary algebraic calculation and reasoning show that one can choose the parameters
α,β and γ in the definition of the simple quadratic map h defined on the real line by

h(x) = αx
2

+ βx + γ (11)

in such a way that h maps [a,b] to [c,d] bijectively and both h and its inverse are continuous hence
(Borel) measurable. Thus (the restriction to [a,b] of) h is a re-labeling between ([a,b],S[a,b],p[a,b])
and ([c,d],S[c,d],p[c,d]). Specifically, the parameters below have this feature

α =
d− c

(b−a)2
(12)

β = −2a
d− c

(b−a)2
(13)

γ = a
2 d− c

(b−a)2
+ c (14)

Furthermore, if � is a real number such that [a,a + �] ⊆ [a,b] then

p[a,b]([a,a + �]) =
�

b−a

and since h takes [a,a + �] into [c,c + d−c
(b−a)2 �

2] one has

p[c,d](h
[
[a,a + �]

]
) =

1

d− c

(
c +

d− c
(b−a)2

�
2
)

It is clear then that for many �

p[a,b]([a,a + �]) =
�

b−a
6=

1

d− c

(
c +

d− c
(b−a)2

�
2
)

= p[c,d](h
[
[a,a + �]

]
) (15)

which is a violation of Labeling Invariance.

7



Remark 2. Note that the above example is typical in the following sense: A probability measure
space is called a standard probability space if X is a complete, separable metric space and S is the Borel
σ algebra of X. Standard, non-atomic probability measure spaces are isomorphic to ([a,b],L[a,b],p[a,b])
with some interval [a,b] where L[a,b] is the algebra of Lebesgue measurable sets in [a,b] (see [1][Chapter
1, p. 3]). Hence the above example gives a large number of re-labelings that violate Labeling
Invariance in the category of spaces (X,S,pH) with X being a complete, separable metric space.
This covers all the spaces that occur in connection with Bertrand’s Paradox.

6 Attempts to save Labeling Invariance

One may attempt to defend Labeling Invariance by trying to block the emergence of the general
Bertrand’s Paradox. The previous section makes it clear what the possible strategies are to achieve
this: One can impose some extra condition on re-labelings that entails either that re-labelings satis-
fying the extra conditions do not exist (Strategy A) or that the re-labelings satisfying the additional
conditions force the re-labelings to be isomorphisms of the probability spaces (Strategy B). Although
not formulated in this terminology, Bangu’s recent attempt [2] is an example of Strategy A. We show
below that Bangu’s suggestion for Strategy A is ambiguous however and that resolving the ambigu-
ity makes it either a trivial case of Strategy B or is unsuccessful. A successful implementation of
Strategy B is to say that it is unreasonable to expect a re-labeling to preserve probabilities unless the
re-labeling also preserves our epistemic status with respect to the elementary events: after all, the
Principle of Indifference states that pH is the empirically correct probability only if the group
structure of X expresses epistemic neutrality. So the following stipulation is in the spirit of the
Principle of Indifference:

Definition: The re-labeling h between probability spaces (X,S,pH) and (X′,S′,p′H) preserves the
epistemic status if it is a group isomorphism between X and X′.

Since the probability measures pH and p
′
H are completely determined by the respective group

actions, re-labelings that preserve the epistemic status are isomorphisms between the measure spaces,
hence no Bertrand’s Paradox can arise with respect to such re-labelings; furthermore, not every re-
labeling is a group isomorphism – thus this strategy works in a non-trivial way.

Bangu’s suggestion is that one should only expect Labeling Invariance to hold for bijections
that “preserve randomness” – this is his Assumption R – Bertrand’s paradox is only a paradox
in his view if Labeling Invariance is violated by re-labelings satisfying the randomness condition,
which, he claims, has not been shown and the burden of proof is on those who claim such re-labelings
exist. It is clear from the wording of his paper that he conjectures that no such proof can be given,
i.e. that no randomness preserving re-labelings exist that violate Labeling Invariance (i.e. that he
is following Strategy A).

As Bangu also points out, the notion of randomness is notoriously both vague and rich: the
adjective “random” can be applied to different entities (events, processes, dynamics, ensembles etc.),
it can come in the form of a pre-theoretical informal intuition, in the form of precise mathematical
definitions, and it also can come in degrees. Thus one has to be very careful and specific when it
comes to the problem of whether “randomness is preserved” under a re-labeling of the elementary
random events. Bangu leaves it deliberately open in what sense precisely “randomness” might not
be invariant under re-labeling of the random events; hence his suggestion remains vague. No matter
what kind of notion of randomness one has in mind, if it is to be relevant for probabilistic modeling of
a phenomenon, then it must be expressible in terms of probabilities, since the basic principle guiding
the modeling of phenomena by probability theory is the maxim:

Distribution Relevance: “A property is probability theoretical if, and only if, it is describable in
terms of a distribution” [14][p. 171].

In the spirit of Distribution Relevance one can take the position that randomness of a phe-
nomenon expressed by “randomness” of the random variables that describe the phenomenon are
encoded in the distribution of the random variables. Consequently, under this interpretation of ran-
domness, if one is given two probability models (X,S,p) and (X′,S′,p′) of a given phenomenon and
h: X → X′ is a re-labeling between (X,S,p) and (X′,S′,p′), then h preserves the randomness of
the two probabilistic descriptions if and only if it holds that if f : X → R is any random variable
in (X,S,p) with distribution p ◦ f−1 then the distribution p′ ◦ f

′−1
in (X′,S′,p′) of the re-named

random variable f′ = f ◦h−1 coincides with p◦f−1:(
p
′ ◦ (f ◦h−1)−1

)
(d) =

(
p◦f−1

)
(d) for all d ∈B(R) (16)

and conversely: for every random variable g′ : X′ → R which is the re-named version of a random
variable g = g′ ◦ h in (X,S,p) it holds that the distribution p′ ◦ g′−1 in (X′,S′,p′) of g′ and the

8



distribution p◦g−1 of g = g′ ◦h in (X,S,p) coincide:

(p◦ (g′ ◦h)−1)(d) = (p′ ◦g′−1)(d) for all d ∈B(R) (17)

Since the random events themselves are random variables, the two equations (16)-(17) must hold for
every characteristic function χA (A ∈S) in place of f and every characteristic function χA′ (A′ ∈S′)
in place of g′ as well, so this requirement of preserving randomness amounts to the demand that the
following two equations hold:

p
′
(h[A]) = p(A) for all A ∈S (18)

p(h
−1

[A
′
]) = p

′
(A
′
) for all A

′ ∈S′ (19)

which is precisely Labeling Invariance (eqs. (6)-(7)). So, if “preserving randomness by re-labeling”
in Assumption R is understood in the spirit of Distribution Relevance as conditions (16)-(17)
then the only randomness-preserving re-labelings are the isomorphisms and no Bertrand paradox can
arise indeed – requiring preserving randomness in this sense is equivalent to the requirement that the
re-labelings are isomorphism. Strategy A, so interpreted, is trivial.

One can try to argue that this is an extremely strong interpretation of “preserving randomness”
and that randomness also can be interpreted differently as expressed by some other property Φ(p)
of the probability measure p. For instance, one has the intuition that a probability measure sharply
concentrated on a single point in X is far less “random”, it represents much more certainty by having
zero variance than a probability distribution that has a large variance. The usual (Shannon) entropy
of a probability measure also can be taken as a measure of “randomness” of the phenomenon that
the probability model describes [4][p. 61-62]. Thus one can interpret the requirement of “preserving
randomness under re-labeling” in Assumption R in different ways depending on what property Φ
one chooses:

Assumption R[Φ]: If (X,S,pH) and (X′,S′,p′H) are two probability spaces and h is a re-labeling
between X and X′ then we say that Assumption R[Φ] is satisfied if both Φ(pH) and Φ(p

′
H) hold.

It is clear then that if there is a property Φ of randomness of a probability measure and there exists
probability spaces (X,S,pH) and (X′,S′,p′H) with a re-labeling h: X → X

′ such that Assumption
R[Φ] is satisfied but Labeling Invariance is violated by h then Bertrand’s paradox re-emerges.

The variance and the entropy are such properties: Consider the probability spaces ([a,b],S[a,b],p[a,b])
and ([c,d],S[c,d],p[c,d]) described in the Example in section 5. The variance σ(p[a,b]) of the normal-
ized Lebesgue measure p[a,b] on any interval [a,b] is by definition equal to

σ(p[a,b]) =

∫ b
a

1

b−a
x
2
dx−

[∫ b
a

1

b−a
xdx

]2
=

(b−a)2

12
(20)

and the entropy E(p[a,b]) of p[a,b] is by definition

E(p[a,b]) = −
∫ b
a

1

b−a
log(

1

b−a
)dx = log(b−a) (21)

It follows then that if b−a = d− c = t then

σ(p[a,b]) = σ(p[c,d]) =
t2

12
(22)

E(p[a,b]) = E(p[c,d]) = log(t) (23)

On the other hand, the map h defined in the Example remains a re-labeling even if b−a = d− c
and Labeling Invariance is violated by this map because for b−a = d−c = t eq. (15) entails that
for many � we have

p[a,b]([a,a + �]) =
�

t
6=

1

t

(
c +

1

t
�
2
)

= p[c,d](h
[
[a,a + �]

]
) (24)

Thus Bertrand’s paradox re-emerges: The probability space ([c,d],S[c,d],p[c,d]) can be regarded as a
re-named version of the probability space ([a,b],S[a,b],p[a,b]) via the re-labeling h defined by (11) and
(12)-(14), furthermore, if b−a = d−c then this re-labeling satisfies Assumption R[Φ] with Φ being
the variance or entropy, and because of (15) h violates Labeling Invariance (6)-(7).

One also can try to question Distribution Relevance. But if one gives up Distribution Rel-
evance and interprets “randomness” in a way that makes randomness not expressible exclusively in
terms of the distributions involved, then the appropriately modified Assumption R constrains even
less the emergence of Bertrand’s Paradox. Rowbottom and Schackle [22] take Assumption R to be
(a technically undefined) “unpredictability” and argue (informally) that there are re-labelings that

9



preserve “unpredictability” and which are not isomorphisms, contrary to what Bangu [2] seems to
conjecture. As a technically more explicit example, assume that a dynamic {αt : t ∈ R} is given on
([a,b],S[a,b],p[a,b]) and a dynamic {α′t : t ∈ R} is given on ([c,d],S[c,d],p[c,d]), where αt and α′t are
one parameter groups of measure preserving maps on [a,b] and [c,d] respectively. As randomness of
the dynamical systems ([a,b],S[a,b],p[a,b]),{αt}) and ([c,d],S[c,d],p[c,d]),{α′t}) one can take the ran-
domness of the respective dynamics such as ergodicity, or mixing, which are not expressible in terms
of p[a,b] and p[c,d] only. Given the re-labeling h between ([a,b],S[a,b],p[a,b]) and ([c,d],S[c,d],p[c,d])
described in the Example in section 5 that violates Labeling Invariance one can then specify the
dynamics {αt} and {α′t} in such a way that they are both ergodic, [4][p. 34], generating a Bertrand’s
Paradox, or in such a way that {αt} is ergodic whereas {α′t} is not, which would be a violation
of preserving randomness (Assumption R) hence not a case of Bertrand’s Paradox (according to
Bangu’s requirement) – anything is possible under such a dynamical interpretation of randomness.

Thus the emergence of Bertrand’s paradox cannot be blocked in a non-trivial way by requiring
the paradoxical examples to satisfy the randomness test and showing that they cannot pass this test:
unless one requires in effect that the re-labeling be an isomorphism, Bertrand’s Paradox emerges: If
Distribution Relevance is accepted and randomness is interpreted as measured by the variance or
entropy of the probability measures, then elementary examples can be given that show violation of
Labeling Invariance. If Distribution Relevance is abandoned, then the randomness requirement
can be satisfied even more easily.

7 Comments on the classical interpretation of probabil-
ity

Probability theory, as specified by the Kolmogorovian axioms, is part of pure mathematics. Just
like other branches of pure mathematics, probability theory also can be used to describe certain
phenomena however. One has to distinguish such applications of probability theory both from pure
mathematics and from interpretations of probability as this latter term is used in philosophy of
science.

In an application of probability theory one relates the mathematical elements in a triplet (X,S,p)
to non-mathematical entities. This involves two tasks:

Event Interpretation To specify what the elements in X and S stand for.
Truth Interpretation To clarify when the proposition “p(A)=r” is true/false.

In an application, probability theory thus becomes a mathematical model of a certain phenomenon
that is external to mathematics. A probability measure space is a good model of the phenomenon if it
has two features: descriptive accuracy and predictive success. Descriptive accuracy means that under
the fixed specification of the Event and Truth Interpretations propositions such as p(A) = r are true
about events that have been observed in the past. Predictive success means that the probabilistic
propositions p(A) = r will be true in future observations. It is clear that both descriptive correctness
and predictive success are robustly empirical features and that descriptive accuracy up until time
T does not entail predictive success for times after T – this is just a particular formulation of the
problem of induction; hence, whether a probability space is a good model is a question that can be
answered only on the basis of empirical considerations. This is of course not new, there is nothing
peculiar or mysterious about probabilistic modeling, probabilistic scientific theories are just like any
scientific theory from this perspective.

The mathematical notion of isomorphism between probability measure spaces is in complete har-
mony with the application of probability theory – and so is the General Bertrand Paradox: The
Event Interpretation and Truth Interpretation are conceptually different issues, the former does not
determine the latter, and, accordingly, two probability spaces are defined to be isomorphic if two
conditions are satisfied: the random events in the two spaces are connected by a re-labeling and the
re-labeling preserves the probabilities. From the perspective of the notion of isomorphisms of proba-
bility spaces finite probability spaces with the uniform probability measure just happen to have the
“contingent” feature that in this category re-labelings are isomorphisms; in this case the re-labelings
contain enough information to make them isomorphisms.

This contingent feature is very deceptive however because it gives the impression that Labeling
Invariance is the proper way to ensure Labeling Irrelevance. It is because of this conflation of
Labeling Invariance and Labeling Irrelevance that violation of Labeling Invariance in the
category of Haar measure spaces (i.e. General Bertrand’s Paradox) appears paradoxical. But there
is nothing paradoxical about this, Labeling Irrelevance is respected in probabilistic modeling
perfectly well – but it is respected not by Labeling Invariance holding true: If the elements in

10



the pair of sets (X,S) label the (elementary, respectively, general) random events of some random
phenomenon, then one is free to use another pair of sets (X′,S′) to label the events as long as no
random events are lost in X′ and S′, i.e. as long as there is a re-labeling h between X and X′.
Labeling Irrelevance says that the choice of (X,S) or (X′,S′) does not affect the probabilities of
the random events and this is in harmony with the fact that fixing either (X,S) or (X′,S′) does
not determine any probability measure on either (X,S) or (X′,S′): any of the (continuum number
of) mathematically possible probability measures can be defined on both (X,S) and (X′,S′). So,
if the probability measure p is such that (X,S,p) is a descriptively accurate probabilistic model of
the phenomenon in question, then (X′,S′,p′) with p′ defined by p′ ≡ p ◦ h also is a descriptively
accurate model of the same phenomenon and (X,S,p) and (X′,S′,p′) are isomorphic with respect
to h. Conversely, if (X,S,p) and (X′,S′,p′) are isomorphic probability spaces with respect to h,
then (up to a measure zero set) h is a re-labeling of the events preserving probabilities, and either
(X,S,p) or (X′,S′,p′) can be used to describe the phenomenon, choosing any of them – choosing
any of the two labelings (X,S) or (X′,S′) in particular – is a matter of convention. In short:
Labeling Irrelevance is encoded in the notion of isomorphism of probability spaces and in the
claim that isomorphic probability spaces can be used to describe the same phenomenon. Note that
this interpretation of how Labeling Irrelevance is ensured in probabilistic modeling does not depend
on any particular features of the probability spaces used in modeling; in particular it is not assumed
that the probability measure is the uniform probability or a Haar measure. Indeed, it would be
unacceptable if Labeling Irrelevance would only hold for situations in which the probabilities of
the events are given by such special measures.

Interpretations of probability are typical classes of applications of probability theory, classes con-
sisting of applications that possess some common features, which the interpretation isolates and
analyzes. The elementary classical interpretation concerns the application of the particular probabil-
ity spaces (Xn,S,pu), where the set Xn of elementary events is finite and the probability measure pu
is the uniform probability measure on Xn. The problem with the classical interpretation (understood
with the amendment of the Principle of Indifference) is not Bertrand’s Paradox, i.e. not that
the Principle of Indifference cannot be consistently generalized from (Xn,S,pu) to the case of an
infinite number of elementary random events: We have seen in section 3 that a General Princi-
ple of Indifference and a General Classical Interpretation can be consistently formulated in
terms of Haar measure spaces (X,S,pH) and one can in principle maintain the General Classical
Interpretation. What one has to abandon is the general version of Labeling Invariance, which
is not maintainable in view of the General Bertrand Paradox. But the Principle of Indifference
and Labeling Invariance are independent ideas: One can reject Labeling Invariance completely,
and one can do so without abandoning the conceptually important Labeling Irrelevance because,
as we have seen above, Labeling Invariance is different from Labeling Irrelevance. One also
can in principle restrict Labeling Invariance to the domain in which it holds: to the category of
probability measure spaces with a finite number of random events or to re-labelings that preserve the
epistemic status. All these options are logically consistent with the logically consistent formulation
of the General Principle of Indifference and with the General Bertrand’s paradox – Bertrand’s
Paradox is defused.

This is not to say that the classical interpretation of probability is acceptable however. The
main problem with the classical interpretation is that it disregards the empirical character of the
applications of probability theory and gives the impression that descriptive accuracy and predictive
success in applications are based on (and can be ensured by referring to) an priori-flavored principle
that expresses some sort of epistemic indifference about random events. But this is not possible, which
is shown by the difficulty (often pointed out in connection with the Principle of Indifference [8])
that it is unclear how to specify the precise content of “epistemic neutrality” in such a way that
the Principle of Indifference does not become circular and holds nevertheless: The Principle of
Indifference holds only if epistemic neutrality does entail that the probabilities of the events given
by the uniform probability measure will be equal to the frequencies of events in actual trials producing
elementary random events, and such a conclusion cannot be validly based on a priori considerations
– if it could, the Principle of Indifference would have solved the problem of induction.

Note in this connection that Jaynes’ [11] resolution of Bertrand’s Paradox cannot be interpreted as
an instance of deducing empirically correct probabilities from abstract principles in the sense which is
relevant for the claims we made above about non-deducibility of empirical frequencies from abstract
principles. The logic of Jaynes’ treatment of Bertrand’s problem is the following:

A Jaynes shows that certain mathematical conditions entail a unique probability distribution on
the two-dimensional plane.

B He argues that those mathematical conditions should be interpreted as expressing symmetry
tacit in the formulation of Bertrand’s problem.

11



C He shows that there exists an empirical arrangement, a particular way of actually generating an
ensemble of events, in which the relative frequencies approximate the probability of the event.
(Jaynes was dropping straws on a circle from a ladder.)

A+B+C cannot be regarded as derivation of empirically correct probabilities from non-empirical
(a priori) conditions in the relevant sense of “derivation” because Jaynes did not show that the
ensemble of events has to be generated the way C claims is possible. That the ensemble in which
the frequencies are equal to the probabilities given by the probability measure Jaynes deduced from
symmetries is produced in a particular way (by dropping straws from a ladder) cannot be deduced in
a logically valid manner from anything a priori because the actual production of the ensemble in this
particular way is a contingent, empirical fact – no conceivable abstract Principle of Indifference
can validly entail that Jaynes dropped straws from a ladder in a specific way. Jaynes just showed
that the mathematical probability theory with the particular probability measure he derived from
mathematical symmetry assumptions can be a descriptively accurate model of a segment of reality
under the frequency interpretation of probability, i.e. that there exist experimentally executable
conditions that generate random events in such a way that the symmetry mentioned in B is in
fact a feature of the experimental arrangement and that frequencies of events in the ensemble are
described by the probability measure he derived. That Jaynes did not derive that this has to be
so is also confirmed by the fact that, as Marinoff [16] shows, one can distinguish different types
of random generators representing different types of randomness and, depending on which random
generator actually produces the random events featuring in a Bertrand Paradox type situation, one
obtains different empirical probability distributions – none of these can be derived from a priori
considerations.

Note finally that it also is possible to interpret the General Bertrand’s Paradox presented in this
paper as an argument in favor of the claim that probability measure spaces with an uncountably
infinite number of random events are too general and abstract to be free of conceptually controversial
features. This conclusion might also be supported by referring to the impossibility of a frequency
interpretation of probabilities in certain probability measure spaces with “too large” sets of random
events and such an interpretation would be in line with the well-known position that rejects countable
additivity as a reasonable feature of probability measures interpreted as representing degrees of belief.
Given that a wide range of probability measure spaces with uncountably many random events have
found very successful applications in the sciences, especially in physics, we would not draw such a
radical conclusion however. We prefer the modest interpretation of the General Bertrand’ Paradox:
Bertrand’s Paradox shows violation of Labeling Invariance, i.e. that re-labelings are not necessarily
isomorphisms in the category of Haar probability measure spaces with infinite random events. But
the violation of Labeling Invariance does not undermine the classical interpretation of probability
understood with the Principle of Indifference, and violation of Labeling Invariance also is
in complete harmony with how mathematical probability theory is used in the sciences to model
phenomena. In particular, violation of Labeling Invariance does not entail violation of Labeling
Irrelevance; yet, irrespective of Bertrand’s Paradox, the content of the Principle of Indifference
cannot be specified in such a way that it can establish the classical interpretation of probability (with
the frequency link) as descriptively accurate or predictively successful.

Appendix

This Appendix recalls some elementary facts about the Haar measure. Standard references for
the Haar measure are [18] and [9][Chapter XI.], for a more recent presentation see [5].

X is called a topological group with multiplication (x,y) 7→ x ·y and inverse x 7→ x−1 if the map
(x,y) 7→ x−1 ·y is continuous (x,y ∈ X). A measure p on the Borel algebra S of the group X is called
left invariant (respectively right invariant) with respect to the group action if eq. (25) (respectively
eq. (26)) below hold

p(A) = p(xA) for all x ∈ X A ∈S (25)
p(A) = p(Ax) for all x ∈ X A ∈S (26)

where for an x ∈ X, the sets xA and Ax are defined by

xA = {x ·y : y ∈ A} (27)
Ax = {y ·x : y ∈ A} (28)

The measure p is called invariant if it is both left and right invariant, i.e. if

p(A) = p(xA) = p(Ax) for all x ∈ X A ∈S (29)

12



On any locally compact topological group there exists both a left pLH and a right p
R
H invariant Haar

measure and they are unique up to multiplication by a constant. The left and right invariant Haar
measures are in general different. Since both the left and Haar measure is unique up to constant
multiplication, and since for any x ∈ X the measure px(A)

.
= pLH(Ax) is again a left invariant

measure, there exists a real number ∆(x) such that px(A) = ∆(x)p
L
H(A). The map x 7→ ∆(x)

is called the modular function of the group. If ∆(x) = 1 for all x, then the groups are called
unimodular; for unimodular groups the left and right invariant Haar measures coincide and yield an
invariant measure. Compact and locally compact abelian groups are unimodular. The Haar measure
is bounded if and only if X is compact – the Haar measure is then a probability measure.

The canonical examples of unbounded Haar measures are the Lebesgue measure on the real line
and the Lebesgue measure on Rn. It is shown below that the normalized restrictions of the Lebesgue
measure on Rn to subsets of the form ×ni [ai,bi) in Rn also can be regarded as Haar measures in
their own right with respect to a compact group G. This entails that the Lebesgue measure on the
closed set ×ni [ai,bi] also can be viewed as a Haar measure with respect to G because the Lebesgue
measure space over ×ni [ai,bi) and over ×ni [ai,bi] are isomorphic. (Note that G is not the shift; it
cannot be since shifted subsets of [0, 1) are not necessarily subsets of [0, 1) and the group of “shifts
modulo 1” do not form a topological group due to discontinuity of the “shift modulo 1” operation.)
Since [0, 1) can be mapped onto [a,b) by a continuous linear bijection connecting the (normalized)
Lebesgue measures on the intervals [0, 1) and [a,b), to see how the Lebesgue measure on [a,b) is a
Haar measure in its own right, it is enough to see how the (normalized) Lebesgue measure p[0,1) on
the interval [0, 1) emerges as a Haar measure. Let

S
1

=
{
z ∈ C : |z| = 1

}
be the unit circle on the complex plane. As S1 is a compact topological subgroup of C with the
multiplication of complex numbers as the group operation, there exists a normalized Haar measure
pH on S

1. The exponential function f defined by

f : [0, 1) → S1, f(t) = e2πit

is a continuous and continuously invertible bijection between the unit interval [0, 1) and the unit circle
S1; hence both f and its inverse are measurable. We claim that f is a measure theoretic isomorphism
between the interval [0, 1) with the Lebesgue measure on it and S1 with the measure pH on it; i.e.
that

pH = p[0,1) ◦f
−1

(30)

To verify (30), by the uniqueness of Haar measures, it is enough to show that p[0,1) ◦f−1 is a Haar
measure, i.e. that p[0,1) ◦ f−1 is invariant with respect to the group operation in S1, which is the
multiplication of complex numbers. Since the exponential function f turns addition of real numbers
into multiplication of complex numbers, for B ⊂ S1 and z ∈ C we have

f
−1

(B ·z) = f[B] + t mod 1 (31)

where the translation
Y 7→ Y + t mod 1 (32)

is the standard shift of set Y ⊂ [0, 1) by t followed by “pulling back” into [0, 1) the part of Y that is
shifted out of the bounds of [0, 1); formally:

Y + t mod 1 =
(
Y ∩ [0, 1 − t) + t

)
∪
(
Y ∩ [1 − t, 1) − (1 − t)

)
p[0,1) is translation invariant on [0, 1) in the sense that for any measurable set A ⊆ [0, 1) and 0 ≤ t < 1
we have

p[0,1)(A) = p[0,1)(A + t mod 1),

so we have

pH(B ·z) = p[0,1)(f
−1

(B ·z)) = p[0,1)(f
−1

(B) + t mod 1) = p[0,1)(f
−1

(B)) = pH(B)

The Lebesgue measure pn[0,1) on the n-dimensional cube [0, 1)
n also can be regarded as a Haar measure:

one can consider the Haar measure pnH on the n-dimensional torus

T
n

= S
1 ×S1 ×···×S1 (n times)

which is a compact topological subgroup of Cn with the coordinate-wise multiplication of complex
numbers as group operation. Put

f : [0, 1)
n → Tn, f(t0, . . . , tn) =

(
e
2πit0, . . . ,e

2πitn
)

13



Then f is a continuous and continuously invertible bijection and, applying the previous argument in
each coordinates, one concludes

p
n
H = p

n
[0,1) ◦f

−1

References

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