Proof of the Ergodic Theorem
- Department of Mathematics, Harvard University
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Communicated December 1, 1931

Let
be a system of n differential equations valid on a closed analytic manifold M, possessing an invariant volume integral, and otherwise subject to the same restrictions as in the preceding note, except that the hypothesis of strong transitivity is no longer made.
We propose to establish first that, without this hypothesis, we have
for all points P of the surface σ save for points of a set of measure 0. In other words, there is a “mean time τ(P), of crossing” of σ for the general trajectory.
The proof of the “ergodic theorem,” that there is a time-probability p that a point P of a general trajectory lies in a given volume v of M, parallels that of the above recurrence theorem, as will be seen.
The important recent work of von Neumann (not yet published) shows only that there is convergence in the mean, so that (1) is not proved by him to hold for any point P, and the time-probability is not established in the usual sense for any trajectory. A direct proof of von Neumann’s results (not yet published) has been obtained by E. Hopf.
Our treatment will be based upon the following lemma: If
then
We consider only the first case, for the proof of the second case is entirely similar. In analogy with the preceding note, define the distinct measurable sets U1, U2, … on Sλ so that for P in Un
The quantity ϵ > 0 is taken arbitrarily. It is, of course, clear that for every point P of Sλ
for infinitely many values of n, so that all such points belong to at least one of the sets U1, U2, …. Now, by the argument of the earlier note, we infer
where
for any ϵ > 0, whence the inequality of the lemma.
The recurrence theorem stated results directly from this lemma. Consider the measurable invariant set of points P on σ for which
for infinitely many values of n (see the preceding note). This is a set Sλ to which the lemma applies. Similarly the set of points P on σ for which
for infinitely many values of n is a set
The set Sλ diminishes and the set
for any ∧, which is absurd. Moreover, when λ tends toward 0, Sλ becomes vacuous, since there is a least time of crossing, λ0. In a similar way,
If then Sλ and
Consider the set of points belonging to
for infinitely many values of n. These form an invariant measurable subset
which are mutually contradictory.
Hence we infer that all of the points P of
for any μ > λ and for n = nP sufficiently large, that is,
Likewise we infer that for all of the points of
It follows then that for points P of
Two such sets
Choose now any two values of λ, say λ,μ with λ < μ, not belonging to this numerable set, and consider the points of Sλ which do not belong to Sμ. These form an invariant measurable set Sλ,μ, such that for any point P of this set
and also
since Sλ,μ is essentially identical with the part of
By choosing a set of values such as λ,μ sufficiently near together we infer then that for all of the points of σ except a set of measure 0, the oscillation of tn(P)/n, as n becomes infinite, is less than an arbitrary δ > 0.
Obviously then the stated recurrence theorem is true.
It should also be noted that if tn/P denotes the time to the nth crossing as time decreases, the same result holds if n tends toward ± ∞, with the same limit except for a set of points P of measure 0. This follows at once from the fact that (8) may be written
where P of Sλ,μ is replaced by Tn(P); and (9) may be given a corresponding form.
This theorem of recurrence admits of certain evident extensions. In the first place there is no need to restrict attention to the analytic case. Moreover, instead of a single surface σ, any measurable set σ∗, imbedded in a numerable set of distinct ordinary surface elements with vcosθ > d > 0, throughout, will serve, in which case t∗(P) denotes the time from P on σ∗ to the first later crossing of σ∗.
In order to prove the “ergodic theorem” we observe first that a set σ∗ can be found which cuts every trajectory except those corresponding to equilibrium and others of total measure 0. This is possible; for a numerable set of distinct ordinary surface elements σ1, σ2, … with vcosθ > d > 0 can be found which cut every trajectory not corresponding to equilibrium. If we define σk as the limit of
where σ12 denotes the set of points P of σ2 not on a trajectory cutting σ1, σ123 denotes the set of points of σ3 not on a trajectory cutting σ1, or σ2, etc., it will have the desired properties.
Now let v denote any “measurable” volume in the manifold M, and let
Hence the same reasoning as before is applicable to show that, except for a set of points P of measure,
where
We conclude that the following “ergodic theorem” holds
For any dynamical system of type (1) there is a definite “time probability” p that any moving point, excepting those of a set of measure will lie in a region v; that is,
will exist, where t denotes total elapsed time measured from a fixed point and
For a strongly transitive system p is, of course, the ratio of the volume of v to V.
Evidently the germ of the above argument is contained in the lemma. The abstract character of this lemma is to be observed, for it shows that the theorem above will extend at once to function space under suitable restrictions.
It is obvious that τ(P) and
where the integral on the left is a Stieltjes integral, m(Sλ) being the measure of Sλ.
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