Mean value theorem for Lebesgue integral

Question

Let $f$ be a mesurable function and $g$ be integrable function, and $\alpha, \beta$ are real numbers such that $\alpha \leq f \leq \beta$ a.e . Prove that there exists a real number $\gamma \in [\alpha, \beta]$ such that $$\int f|g|d \mu = \gamma \int |g|d\mu.$$

I have no idea to proceed the proof. I also searched on the Internet but I got nowhere. Can anyone give me some clear hints? Thanks

Answer 1

Simply note

$$ \alpha \int |g|\,d\mu \leq \int f |g|\,d\mu \leq \beta \int |g|\,d\mu. $$ Why does this imply the claim?