Measure of a set of trajectories

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.

• While this does not address the actual question (which is certainly interesting in and of itself), nonetheless in the context of such quotes discussing ergodic transformations, the intention is that the phrase "almost every trajectory" is an abbreviation of the phrase "the trajectory of almost every point". Commented Oct 28, 2022 at 17:45

$$\newcommand{\T}{\mathscr{T}}$$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.