Invariance from integral duality for a measure-preserving transformation ($\mathbb{E}[XZ]=\mathbb{E}[X(Z\circ T)]\forall Z\text{ bounded}⟹X=X\circ T\text{ a.s.}$)

Let $T$ be a measure-preserving transformation of a probability space and $X$ an integrable random variable. If the displayed duality holds for every bounded random variable $Z$, test it on indicator functions to identify $\mathbb{E}[X\mid T^{-1}(\mathcal{F})]=X\circ T$. The measure-preserving transformation gives the same probability distribution to $X$ and $X\circ T$. The principle of conditional expectation preserving a distribution then forces equality almost surely. This does not require an inverse for $T$ or require $X$ to be square-integrable.