Two theorems on closeness of the set of Laplace-type transformations (Q1332099)

Let \(E\) denote some complete, separable, and metric space, \({\mathfrak B}(E)\) the corresponding Borel \(\sigma\)-algebra and \(M\) the set consisting of all finite, non-negative measures on \({\mathfrak B}(E)\) equipped with some \(\sigma\)-algebra \({\mathfrak M}\) such that all mappings defined by \(m\to m(B)\), \(B\in {\mathfrak B}(E)\), are measurable. Furthermore, let \({\mathfrak L}\) stand for the set of all functions \(\varphi: {\mathfrak B}^ +\to \mathbb{R}_ +\) of the type \(\varphi(g)= \int\exp (\int g dm) P(dm)\), \(g\in {\mathfrak B}^ +\), where \({\mathfrak B}^ +\) is introduced as the set consisting of all functions \(g: E\to \mathbb{R}_ +\), which are measurable and bounded, and where \(P\) stands for some probability measure on \({\mathfrak M}\) and \(\mathbb{R}_ +\) denotes some positive half-line. Main result: \(\varphi\in {\mathfrak L}\) holds true for \(\varphi\) being continuous with respect to the topology of pointwise bounded convergence of \({\mathfrak B}^ +\) and satisfying \(\lim_{n\to\infty} \varphi_ n(g)= \varphi(g)\), \(g\in {\mathfrak B}^ +\), for some sequence \((\varphi_ n)_{n\in \mathbb{N}}\) belonging to \({\mathfrak L}\).