= Invariance from integral duality for a measure-preserving transformation
{title2=$\mathbb E[XZ]=\mathbb E[X(Z\circ T)]\ \forall Z\text{ bounded}\Longrightarrow X=X\circ T\text{ a.s.}$}
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>.
Back to article page
