# How do I prove the general integral function?

If I assume $f$ is any real measurable function and $g$ integrable function. Now, let $\alpha, \beta \in R$ such that $\alpha \leq f \leq \beta$ $a.e.$. I want to prove that there exists $\gamma$ with $\alpha \leq \gamma \leq \beta$ such that $\int f|g|dx = \gamma\int |g|dx$. How I do this? Thank you!

-

$\gamma \int |g|\ dx$ is a continuous function of $\gamma$. –  Robert Israel Jan 28 '13 at 19:22
Suppose $g$ is not zero. Set $$\gamma = \frac{\int f|g|}{\int |g|}.$$
Then prove $\alpha \leq \gamma \leq \beta$.
I think you wanted to say "suppose $\int \lvert g \rvert \neq 0$". –  epimorphic Jan 12 at 0:56