Measure of a set of trajectories

Question

In ergodic systems, the average over the whole space equals the average over the trajectories starting from almost every point $x$.

This is often described saying: "the average over the space is equal to the average over almost every trajectory". Google says that "almost every trajectory" appears more than ten thousend times.

It could be just a shortcut, but, rigorously, we can say "almost every trajectory" if there is a measure of sets of trajectories. It would be a measure of sets of sets of $X$ (sic! I did not repeat "set" by mistake). Does this measure really exist? Can you give a literature reference?

I tried this, very trivial. Given a measure $\mu$ of a space $X$, $\mu : P(X) \to \mathbb{R}$ (where $P(X)$ is the power set of $X$), I can define $\mu' : P[P(X)] \to \mathbb{R}$, such that $\mu'(S)=\mu(T)$ where $T$ is the union of all the elements of $S$. I do not know if it is possible to find a suitable $\sigma$-algebra such that this $\mu'$ is really a measure. Actually, I am more interested in knowing whether a widely-known definition of such a measure exists, rather than in the details, e.g. a literature reference to some popular book on measure theory.

Answer 1

I don't know if this is what you want, because it's a little trivial. But here's an answer. I don't know if this is at all considered in the literature.

$\newcommand{\T}{\mathscr{T}}$Let $(X,\Sigma,\mu;\varphi)$ be a measure preserving dynamical system (ergodic too, if you wish). A trajectory (I know it as "orbit") is a set $T_x:=\{\varphi^n(x):n\in\Bbb N_0\}$ where $x\in X$ is distinguished. Maybe let's call the set of all trajectories $\T$.

There is a bijection $\T\cong X$ through $T_x\sim x$. You can make $\T$ a measure space via this identification: $U\subseteq\T$ is measurable iff. $U$'s identification in $X$ is measurable. Then $\mu$ extends naturally to a measure on $\T$. Under this measuring, indeed (when $\varphi$ is ergodic) the space-average is equal to the average of a typical trajectory, "typical" meaning for almost every trajectory.

This could be seen as a justification for the terminology 'almost every [trajectory]' you are finding on Google.