## Show independence result

Hi

If $X$ is a random variable on some probability space $(\Omega, \mathcal{F},P)$, $A$ is a set in $\mathcal{F}$, and for every Borel subset $B$ of $\mathbb{R}$, we have

$\int_{A} I_{B}(X(\omega)) \, dP(\omega) = P(A) \cdot P(X\in B)$

then we say that $X$ is independent of $A$.

Show that if $X$ is independent of an event $A$ then

$I = \int_{A} g(X(\omega)) \, dP(\omega) = P(A) \cdot E\left[g(X)\right]$.

I tried:

$I = P(A) \cdot \int_{\mathbb{R}} g(x) \, d\mu_{X}(x) = P(A) \cdot E\left[g(X)\right]$, just from the definition given.

Here $\mu_{X}(x)$ denotes the distribution measure.

Help appreciated