Suppose $\mu$ and $\nu$ are finite positive measures on a measuable space

Question

The problem is as follow:

Suppose $\mu$ and $\nu$ are finite positive measures on a measuable space (X, M). Show there is $f\in L^1(X,\mu)$ so that $\int f \ d\mu=\int (1-f) \ d\nu$.

I know that $f\in L^1(X,\mu)$ means $\int f d \mu < \infty $, but not sure how to continue here.

Thanks!

Answer 1

Hint: try constant functions! Let $0<a,b<\infty$ be the total measure of $X$ w.r.t. $\mu, \nu$. Then what is the integral of the constant $c$ function w.r.t. $\mu, \nu$?