Let be a probability space. Consider a measurable function . We will find a sufficient condition under which attains maximum (minimum) in .

**Proposition 1** * Let is convex, bounded, convex in , and closed under -a.e. convergence. *

*
*
- If is concave, and upper semicontinuous in , then attains maximum in .
- If is convex, and lower semicontinuous in , then attains minimum in .

* *

*Proof:*

- Let , and be a maximizing sequence, i.e. . By [Kom67], since is bounded in , there exists and relabelled subsequence such that
By convexity, . Moreover, by closedness of under -a.e. convergence, we have . Thus, . On the other hand, we can see by following inequalities:

This implies for some .

- Second claim follows, if we apply first result to .

### Like this:

Like Loading...