The proof of Lemma 4.3 in [RUDLOFF and KARATZAS 2010] is very succinct. In this below, the proof will be rephrased for an exercise of utilization of weak* topology.
The problem is the following. Let be a reference probability space.
is the space of
-measurable random variables
satisfying
almost surely in
. Let
and
are two given sets of probability measures on
whose elements are all absolutely continuous with respect to
. Define
for some constant
by
[RK10] studies max-min problem
In this below, we will present existence of the optimality in .
Proposition 1 There exists
such that
We need some preliminary terminologies for the preparation of the proof. For short, we use to denote
. We consider the following two layers of function spaces.
- The lower level function space is
endowed with norm
- The upper level function space is its dual space of
, that is,
endowed with weak* topology.
In general, the dual space of is bigger than
. However, since
is a
-finite measure as a probability measure, we can identify that the dual of
is simply
. Recall that, we say
(weak* convergence) in
if
We also denote for any probability measure
,
, and
. Note that
, and
.
Now, we are ready for the proof.
Proof: First we prove step by step with a few small related facts.
-
is a subset of the closed (normed) unit ball
, since for all
.
-
is weak* closed. Because, for every
-
-
, otherwise there exists
and
. But this contradicts to (1) by taking
.
-
- Banach-Alaoglu’s theorem tells us that
is weak* compact, and hence
is also weak* compact as a closed subset of
.
- Fix arbitrary
. Then, the function
is a continuous function under weak* topology, due to (1) by taking
.
- The function
is a upper semicontinuous function under weak* topology.
-
attains maximum for some
since
is upper semicontinuous and
is compact under weak* topology.