Rethinking Boltzmannian Equilibrium

Charlotte Werndl and Roman Frigg
Department of Philosophy, Logic and Scientific Method

London School of Economics
c.s.werndl@lse.ac.uk, r.p.frigg@lse.ac.uk
Forthcoming in: Philosophy of Science

Abstract

Boltzmannian statistical mechanics partitions the phase space into macro-
regions, and the largest of these is identified with equilibrium. What justifies
this identification? Common answers focus on Boltzmann’s combinatorial
argument, the Maxwell-Boltzmann distribution, and maximum entropy con-
siderations. We argue that they fail and present a new answer. We char-
acterise equilibrium as the macrostate in which the system spends most of
its time and prove a new theorem establishing that equilibrium thus defined
corresponds to the largest macroregion. Our derivation is completely gen-
eral in that it does not rely on assumptions about the system’s dynamics or
internal interactions.

1 Introduction

Boltzmannian statistical mechanics (BSM) partitions the phase space of a system
into cells consisting of macroscopically indistinguishable microstates. These cells
correspond to the macrostates, and the largest cell is singled out as the equilibrium
macrostate. The connection is not conceptual: there is nothing in the concept of
equilibrium tying equilibrium to the largest cell. So what justifies the association
of equilibrium with the largest cell?

After introducing BSM (Section 2), we discuss three justificatory strategies
based on Boltzmann’s combinatorial argument, the Maxwell-Boltzmann distribu-
tion, and maximum entropy considerations, respectively. We argue all three fail
because they either suffer from internal difficulties or are restricted to systems with
negligible interparticle forces. This prompts the search for an alternative answer.
In analogy with the standard thermodynamic definition of equilibrium, we charac-
terise equilibrium as the macrostate in which the system spends most of its time.

1



We then present a new mathematical theorem proving that such an equilibrium
macrostate indeed corresponds to the largest cell (Section 4). This result is com-
pletely general in that it does it is not based on any assumptions either about the
system’s dynamics or the nature of interactions within the system.

2 Boltzmannian Statistical Mechanics

Let us briefly introduce BSM.1 Consider a system consisting of n particles which is
isolated from the environment and in a bounded container. The system’s state is
specified by a point x = (q,p) (the microstate) in its 6n-dimensional phase space Γ.
The system’s dynamics is determined by its classical Hamiltonian H(x). Energy is
preserved and therefore the motion is confined to the 6n − 1 dimensional energy
hypersurface ΓE defined by H(x) = E, where E is the energy value. The solutions
of the equations of motion are given by the phase flow φt on ΓE, where φt(x) is the
state into which x ∈ ΓE evolves after t time steps. ΣE is the Lebesgue-σ-algebra
and, intuitively speaking, consists of all subsets of ΓE of interest. Γ is endowed
with the Lebesgue measure µ, which is preserved under φt. This measure can be
restricted to a measure µE on ΓE which is preserved as well and is normalised, i.e.
µE(ΓE) = 1. (ΓE, ΣE,µE,φt) is a measure-preserving dynamical system.

Assume that the system can be characterised by a set {v1, ...,vk} of macro-
variables (k ∈ N). The vi assume values in Vi, and capital letters Vi denote the
values of vi. A particular set of values {V1, ...,Vk} defines a macrostate MV1,...,Vk.
We only write ‘M’ rather than ‘MV1,...,Vk’ if the specific Vi do not matter. A set of
macrostates is complete iff (if and only if) it contains all states a system can be in.

A crucial posit of BSM is supervenience: a system’s microstate uniquely de-
termines its macrostate. Every macrostate M is associated with a macroregion
ΓM consisting of all x ∈ ΓE for which the system is in M. For a complete set of
macrostates the ΓM form a partition of ΓE (they do not overlap and jointly cover
ΓE).

2

The Boltzmann entropy of a macrostate M is SB(M) := kB log[µ(ΓM )] (kB is
the Boltzmann constant). The Boltzmann entropy of a system at time t, SB(t), is
the entropy of the system’s macrostate at t: S

B
(t) := S

B
(Mx(t)), where x(t) is the

system’s microstate at t and Mx(t) is the macrostate supervening on x(t).
We denote the equilibrium macrostate by Meq and its macroregion by ΓMeq. A

crucial aspect of the standard presentation of BSM is that Γeq takes up most of
ΓME . To facilitate the discussion, we introduce the term of β-dominance: ΓMeq is

1For details see Frigg (2008, 103–21).
2For lack of space our focus is on the most common case where macrostates are defined

relative to ΓE. Our arguments generalise to cases when the macrostates are defined relative to
other subsets of Γ.

2



β-dominant iff µ(ΓMeq) ≥ β for β ∈ [1/2, 1]. Often equilibrium is characterised as
a state where β is close to one but nothing in what follows depends on a particular
choice of β.

The characterisation of equilibrium as a β-dominant state goes back to Ehren-
fest and Ehrenfest (1912, 30). While different versions of BSM explain the approach
to equilibrium differently, β-dominance is a key factor in all of them. Those who
favour an explanation based on ergodic theory have to assume that ΓMeq takes
up the majority of ΓE because otherwise the system would not spend most of the
time in ΓMeq (e.g. Frigg and Werndl 2011, 2012). Those who see the approach
to equilibrium as the result of some sort of probabilistic dynamics assume that
ΓMeq takes up most of ΓE because they assign probabilities to macrostates that are
proportional to µ(ΓM ) and equilibrium comes out of as the most likely state only
if the equilibrium macroregion is β-dominant (e.g. Boltzmann 1877). Proponents
of the typicality approach see dominance as the key ingredient in explaining the
approach to equilibrium and sometimes even seem to argue that systems approach
equilibrium because the equilibrium region takes up nearly all of phase space (e.g.
Goldstein and Lebowitz 2004).

We do not aim to adjudicate between these different approaches. Our question
is a more basic one: Why is the equilibrium state β-dominant ?

3 Justificatory Strategies

A look at the literature reveals three justificatory strategies. In practice these are
often pursued side-by-side and seen as providing mutual support to each other. We
will assess each of them and argue that none of them is conclusive.

3.1 The Largest Number of Microstates and the Combina-
torial Argument

The leading idea of the first justificatory strategy is that equilibrium is the macrostate
that is compatible with the largest number of microstates. This strategy is exempli-
fied by Boltzmann’s (1877) combinatorial argument.3 The state of one particle is
determined by a point in its 6-dimensional state space Γµ, and the state of a system
of n identical particles is determined by n points in this space. Since the system is
confined to a finite container and has constant energy E, only a finite part of Γµ is
accessible. Boltzmann partitions the accessible part of Γµ into cells of equal size δω
whose dividing lines run parallel to the position and momentum axes. The result is
a finite partition Ω := {ω1, ...,ωm}, m ∈ N. The cell in which a particle’s state lies

3For details see Frigg (2008) and Uffink (2007).

3



is its coarse-grained microstate. The coarse-grained microstate of the entire gas,
called an arrangement, is given by a specification of the coarse-grained microstate
of each of particle.

The system’s macro-properties depend only on how many particles there are in
each cell and not on which particles these are. A specification of the ‘occupation
number’ of each cell is known as a distribution D = (n1,n2, . . . ,nm) where ni is the
number of particles whose state is in cell ωi. Since m and N are finite, there are
only finitely many distributions D1, . . . ,Dk. Each distribution is compatible with
several arrangements, and the number G(D) of arrangements compatible with a
given distribution D = (n1,n2, . . . ,nm) is

G(D) =
N!

n1!n2! . . . ,nm!
. (1)

Every microstate x of ΓE is associated with exactly one distribution D(x). One
then defines the set ΓD of all x that are associated with a distribution D:

ΓD = {x ∈ ΓE : D(x) = D}. (2)

Since macro-properties are fixed by the distribution, distributions are associated
with macrostates. So we ask: which of the distributions is the equilibrium distribu-
tion? Now Boltzmann’s main idea enters the scene: equilibrium is the macrostate
that is compatible with the largest number of microstates. To determine the equi-
librium distribution, Boltzmann assumed that the energy ei of particle i depends
only on the cell in which it is located. Then the total energy is:

m∑
i=1

niei = E. (3)

He furthermore assumed that the number of cells in Ω is small compared to the
number of particles (allowing him to use Stirling’s formula). With the further
trivial assumption that

∑m
i=1 ni = N, Boltzmann shows that µE(ΓD) is maximal

when
ni = γe

λei, (4)

where γ and λ are parameters which depend on N and E. This is the discrete
version of the Maxwell-Boltzmann distribution. Thus the equilibrium macrostate
corresponds to the Maxwell-Boltzmann distribution.4

4What (4) gives us is the distribution with the largest number of microstates (for the Lebesgue
measure) on the 6N-dimensional shell-like domain ΓES specified by the condition (3). It does not
give us the macroregion of maximal size (i.e., the distribution with the largest measure µE on the
6n−1 dimensional ΓE). As Ehrenfest and Ehrenfest (1959, 30) stress, the (not further justified)
assumption is that the possible distributions and the proportion of the different distributions
would not change if macrostates were instead defined on ΓE.

4



Its ingenuity notwithstanding, the combinatorial argument faces a number of
important problems. The first is that it only applies to systems of non-interacting
particles (Uffink 2007, 976-7). It provides a reasonable approximation for systems
with negligible interparticle forces, but any other system is beyond its scope. SM
ought to be a general theory of matter and so this is a serious limitation.

The second problem is the lack of a conceptual connection between equilibrium
in thermodynamics (TD) and the idea that the equilibrium macrostate is the one
that is compatible with the largest number of microstates. In TD equilibrium is
defined as the state to which isolated systems converge when left to themselves and
which they never leave once they have reached it. This has very little, if anything,
in common with the kind of considerations underlying the combinatorial argument.
This is a problem for anyone who sees BSM as a reductionist enterprise. While
the precise contours of the reduction of TD to SM remain controversial, we are
not aware of any contributors who maintain radical anti-reductionism. Thus the
disconnect between the two notions of equilibrium is a serious problem.

Two replies come to mind. The first points out that since ΓMeq is the largest
subset of ΓE, systems approach equilibrium and spend most of their time in ΓMeq.
This shows that the BSM definition of equilibrium is a good approximation to
the TD definition. This is not true in general. Whether a system spends most of
its time in the β-dominant ΓMeq depends on the dynamics. If, for instance, the
dynamics is the identity function, it is not true that a system out of equilibrium
approaches equilibrium and spends most of its time there. The second reply points
out that we know for independent reasons that the equilibrium distribution is the
Maxwell-Boltzmann distribution. This argument will be discussed in the next
subsection, and our conclusion will be guarded.

Finally, the combinatorial argument (even if successful) shows that the equi-
librium macrostate is larger than any other macrostate. However, as Lavis (2005)
points out, this does not imply that the equilibrium is β-dominant for values of β
close to 1. There may be a large number of smaller macrostates who jointly take
up a large part of ΓE. So the combinatorial argument does in fact not show that
equilibrium is β-dominant.

3.2 The Maxwell-Boltzmann Distribution

According to the next justificatory strategy, a system is in equilibrium when its
particles approximately satisfy the Maxwell-Boltzmann distribution (e.g., Penrose
1989). The Maxwell-Boltzmann distribution f(xµ) specifies the fraction of particles
in the gas whose 6-dimensional position and momentum coordinates xµ = (qµ,pµ)
lie in the infinitesimal interval (xµ, xµ + dxµ):

5



f(xµ) =
χ
V

(qµ) (2πmkT )
−3/2

‖V‖
exp

(
−

p2µ
2mkBT

)
, (5)

where T is the temperature, ‖V‖ is the volume of the container, and χ
V

(x) is 1 if
qµ is in the container and 0 otherwise.

This approach is misguided because the Maxwell-Boltzmann distribution is in
fact the equilibrium distribution only for a limited class of systems, namely for
systems consisting of particles with negligible interparticle forces. For particles with
non-negligible interations different distributions correspond to equilibrium (Gupta
2002). Furthermore, for many simple models such as the Ising model (Baxter 1982)
or the Kac-ring (Lavis 2008) the equilibrium macrostate also does not correspond
to the Maxwell-Boltzmann distribution.

This is no surprise given that the two main derivations of the distribution in
effect assume that particles are non-interacting. Boltzmann’s (1877) derivation is
based on equation (3), the assumption that the total energy is the sum of the energy
of the individual particles. This is true only if the particles are non-interacting,
i.e. for ideal gases. While many expect that the argument also goes through for
dilute gases (where this assumption holds approximately), the argument fails for
non-negligible interactions. In Maxwell’s 1860 derivation (see Uffink 2007) the non-
interaction assumption enters via the postulate that the probability distributions
in different spatial directions can be factorised, which is true only if there is no
interaction between particles.

For these reasons the Maxwell-Boltzmann distribution is the equilibrium dis-
tribution only for a limited class of systems and cannot be taken as a general
definition of equilibrium.

3.3 Maximum Entropy

A third stragey justifies dominance by maximum entropy considerations along the
following lines:5 we know from TD that, if left to itself, a system approaches
equilibrium, and equilibrium is a maximum entropy state. Hence the Boltzmann
entropy of a macrostate SB is maximal in equilibrium. Since SB is a monotonic
function, the macrostate with the largest Boltzmann entropy is also the largest
macrostate, which is the desired conclusion.

There are serious problems with the understanding of TD in this argument as
well as with its implicit reductive claims. First, that a system, when left to itself,
reaches equilibrium where entropy is maximal is often taken to be a consequence of

5This strategy has been mentioned to us in conversation but it is hard to track down in print,
at least in pure form. Albert’s (2000) considerations concerning entropy seem to gesture in the
direction of the third strategy. See Winsberg (2004) for a detailed discussion of Albert’s approach.

6



the Second Law of TD, but it is not. As Brown and Uffink pointed out (2001), that
systems tend to approach equilibrium has to be added as an independent postulate,
which they call the ‘Minus First Law’. But even if TD is amended with the Minus
First Law, the conclusion does not follow. TD does not attribute an entropy to
systems out of equilibrium. Thus, characterising the approach to equilibrium as a
process of entropy increase is meaningless from a TD point of view !

Even if all these issues could be resolved, there still would be a question why
the fact that the TD entropy reaches a maximum in equilibrium would imply
that the same holds for the Boltzmann entropy. To justify this inference, one
would have to assume that the TD entropy reduces to the Boltzmann entropy.
But it is far from clear that this is so. A connection between the TD entropy
and the Boltzmann entropy has been established only for ideal gases, where the
Sackur-Tatrode formula can be derived from BSM which shows that both entropies
have the same functional dependence on thermodynamic state variables. No such
results are known for systems with iteractions. Furthermore, there are well-known
differences between the TD and the Boltzmann entropy. Most importantly, the TD
entropy is extensive while the Boltzmann entropy is not (Ainsworth 2012). But an
extensive concept cannot reduce to a non-extensive concept (at least not without
further qualifications).

For these reasons we conclude that maximum entropy considerations cannot be
used to argue for the β-dominance of the equilibrium state.

4 Rethinking Equilibrium

The failure of standard justificatory strategies prompts the search for an alternative
answer. In this section we propose an alternative definition of equilibrium and
introduce a new mathematical theorem proving that the equilibrium state is β-
dominant.

The above strategies run into difficulties because there is no clear connection
between the TD definition of equilibrium and β-dominance. Our aim is to provide
the missing connection by taking as a point of departure the standard TD definition
of equilibrium and then exploiting supervenience to ‘translate’ this macro definition
into micro language.

The following is a typical TD textbook definition of equilibrium: ‘A thermo-
dynamic system is in equilibrium when none of its thermodynamic properties are
changing with time [...]’ (Reiss 1996, 3) In more detail: equilibrium is the state to
which an isolated system converges when left to its own and which it never leaves
once it has been reached (Callender 2001; Uffink 2001). Equilibrium in TD is
unique in the sense thatf the system always converges toward the same equilibrium
state. This leads to the following definition (the qualification ‘strict’ will become

7



clear later):

Definition 1: Strict TD Equilibrium. Consider an isolated system S
whose macrostates are specified in terms of the macro-variables {v1, ...,vk}.
Assume the S is in state MV1,...,Vk at some initial time t0. If there is
a macrostate MV ∗1 ,...,V ∗k satisfying the following condition, then it is the
equilibrium state of S: For all initial states MV1,...,Vk there exists a time
t∗ such that vi(t) = V

∗
i for all all t ≥ t∗, i = 1, ...,k. We then write

Meq := MV ∗1 ,...,V ∗k .
6

Note that this definition incorporates the Minus First Law of TD.
Rephrasing Definition 1 in the framework of BSM leads to the following defini-

tion:

Definition 2: Strict BSM Equilibrium. Consider the same system S
as above, now described as the measure-preserving dynamical system
(ΓE, ΣE,µE,φt), and let M(x) be the macrostate that supervenes on
microstate x ∈ Γ. Let ΓMV1,...,Vk := {x ∈ ΓE : M(x) = MV1,...,Vk} be
the set of all microstates on which MV1,...,Vk supervenes. If there is a
macrostate MV ∗1 ,...,V ∗k satisfying the following condition, then it is the
strict BSM equilibrium state of S: For all initial states x ∈ ΓE there
exists a time t∗ such that MV1,...,Vk(φt(x)) = MV ∗1 ,...,V ∗k for all t ≥ t

∗,
i = 1, ...,k. We then write Meq := MV ∗1 ,...,V ∗k .

Before reflecting on this definition, we want to add a brief comment about
reductionism. Reductive eliminativists may feel that a definition of equilibrium in
SM that is based on ‘top down translation’ of its namesake in TD undermines the
prospect of a reduction of TD to SM. They would argue that equilibrium has to be
defined in purely mechanical terms, and must then be shown to line up with the
TD definition of equilibrium.7

This point of view is not the only game in town and reduction can be had even
if equilibrium is defined ‘top down’ (as in the above definition). First, whether
the above definition undercuts a reduction depends on one’s concept of reduction.
For someone with a broadly Nagelian perspective there is no problem: the above
definition provides a bridge law, which allows the derivation of the requisite macro
regularities from the laws of the micro theory. And a similar argument can be
made in the framework of New Wave Reductionism. Second, equilibrium is a
macro concept: when describing a system as being in equilibrium, we look at it in

6If one wants to avoid the t∗-dependence on the initial state, one can instead demand that
there exists a time t∗ such that vi(t) = V

∗
i for all initial states MV1,...,Vk and all t ≥ t

∗, i = 1, ...,k.
7For a discussion see Dizadji-Bahmani et al. 2010.

8



terms of macro properties. From a micro point of view there are only molecules
bouncing around. They always bounce – there is no such thing as a relaxation of
particle motion to an immutable state. Hence a definition of equilibrium in macro
terms is no heresy.

Definition 2 is too rigid for two reasons. The first reason is Poincaré recurrence:
as long as the ‘M’ in SM refers to a mechanical theory that conserves phase volume
(and there is widespread consensus that this is the case),8 any attempt to justify
an approach to strict equilibrium in mechanical terms is doomed to failure. The
system will at some point return arbitrarily close to its initial condition, violating
strict equilibrium (Frigg 2008; Uffink 2007). The second reason is that such a
justification is not only unattainable but also undesirable. Experimental results
show that equilibrium is not the immutable state that classical TD presents us with
because systems exhibit fluctuations away from equilibrium (Wang et al. 2002).
Thus strict equilibrium is actually unphysical.

Consequently, strict definitions of equilibrium are undesirable both for theoret-
ical and experimental reasons. So let us relax the condition that a system has to
remain in equilibrium for all t ≥ t∗ by the weaker condition that it has has to be
in equilibrium most of the time:

Definition 3: BSM α-Equilibrium. Consider the same system as in
Definition 2. Let fM,x(t) be the fraction of time of the interval [t0, t0 +t]
in which the system’s state is in M when starting in initial state x at t0,
and let α be a real number in [0.5, 1]. If there is a macrostate MV ∗1 ,...,V ∗k
satisfying the following condition, then it is the α-equilibrium state of
S: For all initial states x ∈ ΓE, fMV ∗

1
,...,V ∗

k
,x(t) ≥ α in the limit t →∞.

We then write Mα-eq := MV ∗1 ,...,V ∗k .

An obvious question concerns the value of α. Often the assumption seems to be
that α is close to one. This is reasonable but not the only possible choice. But
for our purposes nothing hangs on a particular choice of α and so we leave it open
what the best choice would be.

One last step is needed to arrive at the definition of equilibrium suitable for
BSM. It has been pointed out variously that in SM, unlike in TD, we should not
expect every initial condition to approach equilibrium (see, for instance, Callender
2001). Indeed, it is reasonable to allow for a set of very small measure ε for which
the system does not approach equilibrium:

Definition 4: BSM α-ε-Equilibrium. Let S and fM,x(t) be as above. Let
α be a real number in [0.5, 1] and let 1 > ε ≥ 0 be a small real number,
and let Y be a subset of ΓE such that µE(Y ) ≥ 1 − ε. If there is a

8Hamiltonian Mechanics falls within this class, but the class is much wider.

9



macrostate MV ∗1 ,...,V ∗k satisfying the following condition, then it is the
α-ε-equilibrium state of S: For all initial states x ∈ Y fMV ∗

1
,...,V ∗

k
,x(t) ≥ α

in the limit t →∞. We then write Mα-ε-eq := MV ∗1 ,...,V ∗k .

Let us introduce the characteristic function of ΓM , 1M (x): 1M (x) = 1 for
x ∈ ΓM and 0 otherwise. Definition 4 implies that for all x ∈ Y :9

lim
T→∞

1

T

∫ T
0

1Mα-ε-eq(φt(x))dt ≥ α. (6)

An important assumption in this characterisation of equilibrium is that µE (and
not some other measure) is the relevant measure. It is often argued that µE can be
interpreted as a probability or typicality measure (Frigg and Hoefer 2013; Werndl
2013). The condition then says that the system’s state spends more than a fraction
α of its time in equilibrium with probability 1−ε, or that typical initial conditions
lie on trajectories which spend more than α of their time in equilibrium.

We contend that the relevant notion of equilibrium in BSM is α-ε-equilibrium.
The central question then becomes: why is the α-ε-equilibrium state β-dominant?
Definition 4 in no way prejudges this question: it says nothing about the size of
ΓMα-eq nor does it in an obvious sense imply anything about it.

That ΓMα-eq is β-prevalent follows from the following theorem, which we prove
in the Appendix:

Equilibrium Theorem: If ΓMα-ε-eq is an α-ε-equilibrium of system S, then
µ(ΓMα-ε-eq) ≥ α(1 −ε).

We emphasise that the theorem is completely general in that no dynamical
assumption is made (in particular it is not assumed that the system is ergodic). So
the theorem also applies to strongly interacting systems such as solids and liquids.

The Equilibrium Theorem is the centre piece of our account. It shows in full
generality that if the system S has an α-�-equilibrium, then the equilibrium state
is β-dominant for β ≥ α(1 − ε).10 This provides the sought-after justification of
the β-dominance of the equilibrium state.

The Equilibrium Theorem makes the conditional claim that if there is an α-
�-equilibrium, then µ(ΓMα-ε-eq) ≥ α(1 − ε). As with all conditionals, the crucial
and often vexing question is whether, and under what conditions, the antecedent
holds. Some systems do not have equilibria. For instance, if the dynamics is
given by the identity function then no approach to equilibrium takes place, and
the antecedent of the conditional is wrong. By contrast, epsilon-ergodicity allows

9This shows that Defintion 4 is closely related to Lavis’ (2005, 255) characterisation of TD-
likeness.

10It is assumed that ε is small enough so that α(1 −ε) ≥ 0.5.

10



for an equilibrium state to exist (Frigg and Werndl 2011). This raises the question
under which circumstances the antecedent is true, which is an important question
for future research.

5 Conclusion

Boltzmannian statistical mechanics partitions the phase space of a system into cells
of macroscopically indistinguishable microstates. These cells are associated with
the system’s macrostates, and the largest cell is identified with equilibrium. What
justifies the association of equilibrium with the largest cell? We discussed three
broad justificatory strategies that can be found in the literature: that equilibrium
is the macrostate compatible with the largest number of microstates, that equilib-
rium corresponds to the Maxwell-Boltzmann distribution and that most states are
characterised by that distribution, and that equilibrium is the maximum entropy
state. We argued that none of them is successful. This prompted the search for
an alternative answer. We characterised equilibrium as the macrostate in which
the system spends most of its time and presented a new mathematical theorem
proving that such an equilibrium state indeed corresponds to the largest cell. This
result is completely general in that it is not based on any assumptions either about
the system’s dynamics or the nature of interactions within the system. It therefore
provides the first fully general justification of the claim that the equilibrium state
takes up most of the accessible part of the system’s phase space.

11



Appendix: Proof of the Equilibrium Theorem

The proof appeals to the ergodic decomposition theorem (cf. Petersen 1983, 81),
stating that for a dynamical system (ΓE, ΣE,µE,φt) the set ΓE is the disjoint union
of sets Xω, each equipped with a σ-algebra ΣXω and a probability measure µω, and
φt acts ergodically on each (Xω, ΣXω,µω). The indexing set is also a probability
space (Ω, ΣΩ,P), and for any square integrable function f it holds that:∫

ΓE

fdµE =

∫
Ω

∫
Xω

fdµωdP. (7)

Application of the ergodic decomposition theorem for f = 1Mα-ε-eq(x) yields:

µE(ΓMα-ε-eq) =

∫
ΓE

1Mα-ε-eq(x)dµE =

∫
Ω

∫
Xω

1Mα-ε-eq(x)dµωdP. (8)

For an ergodic system (Xω, ΣXω,µω,φt) the long-run time average equals the phase
average. Hence for almost all x ∈ Xω:

lim
T→∞

1

T

∫ T
0

1Mα-ε-eq(φt(x))dt =

∫
Xω

1Mα-ε-eq(x)dµω = µω(ΓMα-ε-eq ∩Xω). (9)

From requirement (6) and because φt acts ergodically on each (Xω, ΣXω,µω), for
almost all x ∈ Xω, Xω ⊆ Y :

α ≤ lim
T→∞

1

T

∫ T
0

1Mα-ε-eq(φt(x))dt =

∫
Xω

1Mα-ε-eq(x)dµω. (10)

Let us first consider the case ε = 0, i.e. µE(Y ) = 1. Here from equation (8):

µE(ΓMα-ε-eq) ≥
∫

Ω

αdP = α. (11)

Hence if � ≥ 0, it follows from equation (8) that:

µE(ΓMα-ε-eq) ≥ α(1 −ε). (12)

12



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