My question is in one version of ergodic thrm, we have probability space (R, B, Q) on which a real-valued random variable Y is defined. We further define a sequence of such random variable Y1, Y2,... as $Y_i(x) = Y(T^{i-1} x), x \in R$.
The conclusion is if $E(Y) < \infty$, the $1/n \sum_i ^{n} Y_i $ converges a.s. in mean to E[Y | V], where V is the set of all invariant events.
When we try to understand the relation between this version of ergodic thrm with the most elementary version for strongly stationary processes, textbooks always define $Y: R \rightarrow R$ by $Y(x) = x_1$, or $Y_n(x) = x_n $.
I'm not completely sure why does the Y has to be defined in such an identity form. Will the connection between different versions of ergodic theorem retained if we define the Y in other form?