How do I prove the general integral function?

Question

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!

Answer 1

Hint: intermediate value theorem.

Answer 2

Suppose $g$ is not zero. Set $$ \gamma = \frac{\int f|g|}{\int |g|}.$$

Then prove $\alpha \leq \gamma \leq \beta$.