Let $M$ a metric space, $f:M\to M$ a measurable transformation and $\mu$ a measure on $M$. Prove that if $\int \phi \,d\mu = \int \phi \circ f\, d\mu$ for every bounded continuous function $\phi: M \to \mathbb{R}$, then $f$ is measure preserving transformation.
I know how to do the cases where $\phi$ is measurable and a $L^{1}$ function. But I don't know this case. Any tips?