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.
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.