In Walters' An Introduction to Ergodic Theory on page 34 the Birkhoff Ergodic Theorem is given as follows:
Suppose $T\colon (X,\mathfrak{B},m)\to (X,\mathfrak{B},m)$ is measure-preserving (where we allow $(X,\mathfrak{B},m)$ to be $\sigma$-finite) and $f\in L^1(m)$. Then $(1/n)\sum_{i=0}^{n-1}f(T^i(x))$ converges a.e. to a function $f^*\in L^1(m)$. Also $f^*\circ T=f^*$ a.e. and if $m(X)<\infty$, then $\int f^*\, dm=\int f\, dm$.
Then (before proving the theorem), Walters gives some remarks to this, namely:
If $T$ is ergodic then $f^*$ is constant a.e. and so if $m(X)<\infty$ $f^*=(1/m(X))\int f\, dm$ a.e. If $(X,\mathfrak{B},m)$ is a probability space and $T$ is ergodic we have $\forall f\in L^1(m)\lim_{n\to\infty}(1/n)\sum_{i=0}^{n-1}f(T^i(x))=\int f\, dm$ a.e.
So far so good. I do understand this.
Then some applications are given. And there is one application I do not understand right now. Namely:
Let $T$ be a measure-preserving transformation of the probability space $(X,\mathfrak{B},m)$ and let $f\in L^1(m)$. We define the time mean of $f$ at $x$ to be
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}f(T^i(x))
$$
if the limit exists. The phase or space mean of $f$ is defined to be
$$
\int_X f(x)\, dm.
$$
The ergodic theorem implies these means are equal a.e. for all $f\in L^1(m)$ iff $T$ is ergodic. Since these two means areequated in some arguments in statistical mechanics it is important to verify ergodicity for certain transformations arising in physics. That application to time means and space means is more realistic in the case of a 1-parameter flow $\left\{T_t\right\}_{t\in\mathbb{R}}$ of measure-preserving transformations. The ergodic theorem then asserts
$$
\lim_{T\to\infty}(1/T)\int_0^T f(T_tx)\, dt~~~~~~~(*)
$$
exists a.e. for $f\in L^1(m)$ and equals
$$
\int_X f\, dm
$$
if the flow $\left\{T_t\right\}$ is ergodic and $(X,\mathfrak{B},m)$ is a probability space.
I have two questions to this cited application.
1.) Why is $\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}f(T^i(x))$ called the time mean of $f$ at $x$? And why is $\int_X f(x)\, dm$ called the phase or space mean of $f$?
2.) I do not see why the above cited ergodic theorem asserts (*). Can you please explain that to me? I do not know how Walters could mean that. I am totally helpless…
Best Answer
Let's see if I got it.
Because there is apperearing the Riemann-integral $\int_0^T f(T_tx)\, dt$ I think it is indeed okay to assume that $t\mapsto f\circ T_t$ is Riemann-integrable.
So one way to write the Riemann-integral is $$ \int_0^T f(T_tx)\, dt=\lim_{\Delta_P\to 0}\sum_{k=1}^{m}f(T_{\tau_k}x)(t_k-t_{k-1}), $$ where $0=t_0<\ldots <t_m=T$ is a partition of the intervall $[0,T]$, $\tau_k\in [t_k-t_{k-1}]$ and $\Delta_P:=\max_{k=1,m}\lvert t_k-t_{k-1}\rvert$.
(By the way is not the best idea to use $T$ for the map and the integer but now I stick to that.)
Another way to write the Riemann-integral is to choose the special partition $$ t_0=0, t_k=\frac{Tk}{m}, 1\leq k\leq m $$ to choose $\tau_k=t_k, 1\leq k\leq m$ (so $\Delta_P=\frac{T}{m}$) and then considering the limes $m\to 0$. This is the same.
So consider $$ \int_0^{T}f(T_tx)\, dt=\lim_{m\to\infty}\sum_{k=1}^{m}f(T_{Tk/m}x)\underbrace{(t_k-t_{k-1})}_{=T/m}=\lim_{m\to\infty}\frac{T}{m}\sum_{k=1}^{m}f(T_{Tk/m}x) $$
We can write this as $$ \lim_{m\to\infty}\frac{T}{m}\sum_{k=1}^{m}f(T_{Tk/m}x)=\lim_{m\to\infty}\frac{T}{m}\sum_{k=1}^{m}f(T^k_{T/m}x)=\lim_{m\to\infty}\frac{T}{m}\left(\sum_{k=0}^{m-1}f(T^k_{T/m}x)+f(T^m_{T/m}x)-f(x)\right)\\=\lim_{m\to\infty}\frac{T}{m}\sum_{k=0}^{m-1}f(T^k_{T/m}x)+\underbrace{\lim_{m\to\infty}\frac{Tf(T^m_{T/m}x)}{m}}_{=0}+\underbrace{\lim_{m\to\infty}\frac{Tf(x)}{m}}_{=0}\\=T\lim_{m\to\infty}\frac{1}{m}\sum_{k=0}^{m-1}f(T^k_{T/m}x)=Tf^*(x)\text{a.e.} $$ after the cited ergodic theorem. So it is $$ \lim_{T\to\infty}\frac{1}{T}\int_{0}^T f(T_tx)\, dt=\lim_{T\to\infty}\frac{1}{T}Tf^*(x)=\lim_{T\to\infty}f^*(x)=f^*(x)\text{ a.e.}, $$ i.e. the limits exists a.e. as Walters says.
Am I right?
With greetings and many thanks for your help.