Suppose that $(X,\Sigma)$ is a measurable space, $T$ is a measurable map from $(X,\Sigma)$ to itself, and $\mu$ is a $T$-invariant measure. Define the dynamical system $$ f_n\triangleq T\circ f_{n-1} ; \qquad f_0\triangleq g; \qquad (\forall n \in \mathbb{Z}^+$), $$ where $g:X \rightarrow X$ is $\Sigma$-measurable.
It seems direct from the definition of invariant measure, but just incase I'm missing something: is it true that $$ \int_{x \in X} f_n(x) \mu(dx) = \int_{x \in X} g(x) \mu(dx) ;\qquad (\forall n \in \mathbb{N}) ? $$