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

Af(x)=supnAnf(x)(fL1(m),xX)
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:

limn1nk=0n1Ukf=Pf,()
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,

Uf=fUf=f(f)

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)

(IU)f=fUf=f22re<f,Uf>+Uf22f22re<Uf,f>=2f22re<f,f>=0,
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

=ker(IU)ran(IU)()
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.

References

[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.