Notes for "Kullback-Leibler approximation for probability measures on infinite dimensional spaces"

Article Information: F. J. Pinski, G. Simpson, A. M. Stuart, and H. Weber, SIAM Journal of Mathematical Analysis, 47(6), 4091-4122

Proof of Corollary 2.2:
Since for some probability measure , we know that there exists a sequence such that

Noticing the second statement in Proposition 2.1, the sequence must contain a weak convergence subsequence. We still denote the subsequence as and assume the convergent measure as . From the first statement in Proposition 2.1, we have

Hence, the proof is completed.

Notes for the Proof of Proposition 2.1:
The lower semicontinuity of can be seen from the following formula

Notes for the Proof of Lemma 3.3

As has form domain , the operator
is bounded on by the closed graph theorem

Proof: Let in , and as , we need to prove .
The key point is the following

where we used the condition has form domain .

Notes for Lemma 3.19:
In the proof of Lemma 3.19, the last few lines

for any .......

By my understanding, these statements should be modified as follow
for any .......

Notes for the last part of Appendix B:
How to obtain
Proof: Since

we find that

Then, we obtain . Hence, naturally, we finally arrive at the conclusion.