Intermediate value theorem for integral over measurable space

Question

In the context of a measure space $(X,M, μ)$, suppose $f$ is a bounded measurable function with $a \leq f(x) \leq b$ for $\mu$-a.e. $x \in X$. Prove that for each integrable function $g$, there exists a number $c \in [a, b]$ such that $\int_X f|g| d\mu = c\int_X |g| d\mu$

I tried to use the Intermediate value theorem for integral of Riemann but i had no idea. Somebody have any tip?

Answer 1

You don't need measure theory is an easy use of intermediate theorem for real valued functions…

Consider $$h(c) = c\int_X |g| d\mu - \int_X f|g| d\mu$$

Then $h(a) \le 0, h(b) \ge 0$, hence there is a $c\in[a,b]$ s.t. $h(c) = 0$ what gives you the claim.