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
.