Let $A\ \in \ B(R)$ and $h:\ R\ \rightarrow R$ be a Borel measurable function such that, for any $x\in A^{c}$ , $h(x)\leq 0$ $x\in A$ , $h(x)\leq 1$. Prove that,for any random variable $X$ if $E\left[ h(x)\right]$ exists then: $$E\left[ h(x)\right] \ \leq \ P\ (\ X\ \in \ A\ )\ $$
I know that if $A\ \in \ B(R)$ then $A^{c}\ \in \ B(R)$ thus making $x\ \in \ B(R)$ . But I don't know how to go from there.
Can someone help?