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# Proof of the Ergodic Theorem

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 *σ*, which is invariant under *T*, except possibly for a set of measure 0, and if for any point *P* of this set

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 *U*_{1}, *U*_{2}, … on *S*_{λ} so that for *P* in *U*_{n}

The quantity ϵ > 0 is taken arbitrarily. It is, of course, clear that for every point *P* of *S*_{λ}

for infinitely many values of *n*, so that all such points belong to at least one of the sets *U*_{1}, *U*_{2}, …. Now, by the argument of the earlier note, we infer

where *k*, a measurable part of the invariant set *S*_{λ} and increases toward a limit *U*_{1} + *U*_{2} + … which contains every point of *S*_{λ}. Consequently we obtain by a limiting process

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 *S*_{λ} 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 *S*_{λ} diminishes and the set *σ*, and both sets taken together exhaust *σ*. The measure of the set *S*_{λ} must tend toward 0 as *λ* increases. Otherwise it would tend toward an invariant measurable set of positive measure, S∗, for which the inequality of the lemma holds for λ = ∧, an arbitrarily large positive quantity, and we should infer

for any ∧, which is absurd. Moreover, when *λ* tends toward 0, *S*_{λ} becomes vacuous, since there is a least time of crossing, *λ*_{0}. In a similar way, *λ* from a set of zero measure for *λ* < *λ*_{0} toward the set *σ*.

If then *S*_{λ} and *σ*, one decreasing, the other increasing, they must, for certain values of *λ*, have a common measurable component *T*.

Consider the set of points belonging to

for infinitely many values of *n*. These form an invariant measurable subset *μ*. Otherwise the inequalities of the lemma would give us simultaneously

which are mutually contradictory.

Hence we infer that all of the points *P* of

for any *μ* > *λ* and for *n* = *n*_{P} 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 *λ*’s are evidently distinct except for a set of measure 0. Hence there can exist only a numerable set *λ*_{i} of *λ*, *S*_{λ} are complementary parts of *σ* aside from a set of measure 0.

Choose now any two values of *λ*, say *λ*,*μ* with *λ* < *μ*, not belonging to this numerable set, and consider the points of *S*_{λ} which do not belong to *S*_{μ}. These form an invariant measurable set *S*_{λ,μ}, such that for any point *P* of this set

and also

since *S*_{λ,μ} is essentially identical with the part of *S*_{λ}. We infer then that *t*_{n}(*P*)/*n* oscillates between *λ* and *μ* as *n* tends toward ∞, for all points *P* of *S*_{λ,σ} except a set of measure 0.

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 *t*_{n}(*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 *t*_{n}/*P* denotes the time to the *n*th 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 *S*_{λ,μ} is replaced by *T*^{n}(*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 *v*cos*θ* > *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 *σ*_{1}, *σ*_{2}, … with *v*cos*θ* > *d* > 0 can be found which cut every trajectory not corresponding to equilibrium. If we define *σ*_{k} as the limit of

where *σ*_{12} denotes the set of points *P* of *σ*_{2} not on a trajectory cutting *σ*_{1}, *σ*_{123} denotes the set of points of *σ*_{3} not on a trajectory cutting *σ*_{1}, or *σ*_{2}, etc., it will have the desired properties.

Now let *v* denote any “measurable” volume in the manifold *M*, and let *P* on such a set *σ*∗ lies in *v* before the point *T*(*P*) of *σ*∗ is reached. Thus *t*(*P*)

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* * the elapsed time in v.*

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*(*S*_{λ}) being the measure of *S*_{λ}.