Weak-star convergence in the dual of the continuous functions on the $n$-cube, $1\leq n\leq \infty$

Abstract:

Let $n$ be a positive integer and let $J = \times _{j = 1}^n{[0,1]_j}$ denote the $n$-cube. Let $\mathbf {C} = \mathbf {C}(J)$ denote the (sup norm) space of continuous (real-valued) functions defined on $J$, and let $\mathfrak {M}$ denote the (variation norm) space of (real-valued) signed Borel measures defined on the Borel subsets of $J$. Let $\left \langle {{\mu _l}} \right \rangle$ be a sequence of elements of $\mathfrak {M}$. Necessary and sufficient conditions are given in order that ${\text {li}}{{\text {m}}_l}\int f d{\mu _l}$ exists for every $f \in \mathbf {C}$. After considering a finite dimensional case, the infinite dimensional case is entertained.