Let be a probability space. Consider a measurable function . We will find a sufficient condition under which attains maximum (minimum) in .
- If is concave, and upper semicontinuous in , then attains maximum in .
- If is convex, and lower semicontinuous in , then attains minimum in .
- 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 .