Loic Dubois found the following mistake in my text:

Corollaire 11.4 is wrong, because the probability measures are not an open subset of a topological vector space,
hence the quoted result on convex functions does not apply.
Stronger conditions are needed to ensure the continuity of the Cramer transform.

The consequences for the rest of the text are the following:
Proposition 11.6 as such is not correct, different hypothesis are needed.
In the beginning of the proof of theorem 15.3, one should not rely on Corollary 11.4,
instead one should condition with a compact set and proceed as in the proof of theorem 26.2,  page 138.