The Birkhoff & Maximal Ergodic Theorems

Joel Shapiro

(Revised 11/16/2018)

1. Review


2. The Birkhoff (pointwise) Ergodic Theorem


3. Maximal Averages

Instead of trying to prove directly that the limit the averages Anf(x) exits, we focus on the supremum

of these averages, which always exists (possibly =). We call Af the maximal function of f. The key to proving Birkhoff's Theorem lies with


4. A Consequence of the "Maximal Ergodic Consequence"

Let 𝒢 denote the set of functions in fL1(m) for which limnAnf exists a.e..


5. The Mean Ergodic Theorem

The setting now shifts to a Hilbert space on which acts a contraction U, i.e., a linear transformation for which Uff for each f.

The Mean Ergodic Theorem. Suppose U is a contraction of a Hilbert space . Denote by 𝒦 the null space of IU, and by P the orthogonal projection of onto 𝒦. Then for each f:

the convergence taking place in the norm topology of .

For isometries U this result was published in 1932 by von Neumann [vN].

Comparison with Birkhoff's Theorem. In the setting of Birkhoff's Ergodic Theorem: (X,,m) is a measure space with m(X)<, and a measure-preserving transformation T:XX is used to induce an isometry U on L1(m) by setting Uf=fT for each fL1(m). Now L2(m) is contained in L1(m) since m(X)<, and the measure-preserving-ness of T, which guaranteed that U is an isometry of L1(m), also guarantees that it's an isometry of L2(m). Since L2(m) is a Hilbert space, the Mean Ergodic Theorem shows that () holds in the L2(m)-norm for every fL2(m).


6. Some Hilbert-space preliminaries

Notation. We'll denote the null space of a linear transformation L by "kerL".

A Contraction Theorem. If U is a (linear) contraction on a Hilbert space , then ker(IU)=ker(IU), i.e,


Proof. Since the norm of a (bounded) Hilbert-space operator equals the norm of its adjoint, U is also a contraction.

Suppose fker(IU), i.e., that Uf=f. Then (assuming complex scalars for our Hilbert space)

where in the third line we've used the fact that Uff (since U is also a contraxtion), and in the fourth one the assumption that Uf=f. Thus fker(IU).

The argument so far shows that ker(IU)ker(IU). The reverse inclusion follows upon substuting U for U and using the fact that U=U.

A Contraction Corollary. If U is a contraction then

where the overline denotes "norm-closure of ."

Proof. This follows from the general fact that if L is a bounded operator on then kerL=(ranL). In our case, L=IU, so by the above Theorem, kerL=ranL, from which follows .


7. Proof of the Mean Ergodic Theorem.

We're given a contraction U of Hilbert space . For the operator IU, let 𝒦 denote its null space and its range, i.e, 𝒦={f:Uf=f} and =(IU).


8. Proof of Birkhoff's Ergodic Theorem

We're back to the setting of a measure space (X,,m) with μ(X)<, and measure-preserving transformation T on X, with its induced isometry U on L1(m) defined by Uf=fT.

9. The Lebesgue Differentiation Theorem.


10. Banach's Principle

The method behind the work just done generalizes considerably. Suppose that B is a Banach space and (X,,m) a measure space with m(X)<. Let L0(m) denote the space of (m-equivalence classes of) -measurable, real-valued functions that take finite values a.e.


[Ban] Stefan Banach, Sur la convergence presque partout de fonctionelles linéaires, Bull. Sci. Math., (2) 50 (1926) 27-32 & 36-43.

[Bir] George D. Birkhoff, Proof of the Ergodic theorem, Proc. Nat. Acad. Sci. 17 (1931) 656-660.

[Gar] Adriano Garsia, Topics in Almost Everywhere Convergence, Lectures in Advanced Mathematics #4, Markham Publishing Co., Chicago, 1970.

[Ob] Peter Oberly, The Pointwise Ergodic Theorem and its applications, Lecture Notes, Portland State University Analysis Seminar, November 2018.

[Sh] Joel H. Shapiro, Almost-everywhere convergence ... done right!  Lecture Notes, Portland State University Analysis Seminar, October 2017.

[vN] John von Neumann Proof of the Quasi-Ergodic Hypothesis, Proc. Nat. Acad. Sci. 18 (1932) 70-82.