Suppose I want to solve the following problem
$$\max_{f(\cdot)} \int f(x) g(x) d\mu(x)$$
where the maximization is over measurable functions $f:X\to [0,1]$ , $\mu$ is a finite measure and $g$ is measurable.
Can someone come up with an example such that the problem does NOT have a solution? I imagine one can try to maximize point by point and obtain something that is not measurable and then the problem would not have a solution?
If $g$ is not integrable the problem can be unbounded: Let $\mu$ be the Cauchy distribution on the real line. Let $g(x) = |x|$. Then, we can take $f(x) = 1$ for all $x$ and we have $\int f g d\mu = \int |x| d\mu(x) = \infty$.