Some results on analytic spaces and semi-analytic functions with regard to gambling theory

In gambling theory as well as in dynamic programming, fundamental results are often proved by using properties of the weak topology on the space of real-valued, non-negative Borel measures on some analytic space (e.g., cf. Hinderer 3], Rieder 6], and Blackwell et al.1]). In all these papers, the definition of weak topology is classically based on continuous functions. Here we point out that some of these results can be considerably generalized by using Topsøfe's definition of weak topology based on semi-continuous functions. Furthermore, it turns out that, following this approach, our proofs are much simpler than those in the papers cited above.