Non-archimedean function spaces and the Lebesgue dominated convergence theorem

Abstract

Let $M(X,\mathbb{K})$ be the non-archimedean Banach space of all additive and bounded $\mathbb{K}$-valued measures on the ring of all clopen subsets of a zero-dimensional compact space $X$, where $\mathbb{K}$ is a non-archimedean non-trivially valued complete field. It is known that $M(X,\mathbb{K})$ is isometrically isomorphic to the dual of the Banach space $C(X,\mathbb{K})$ of all continuous $\mathbb{K}$-valued maps on $X$ with the sup-norm topology. Does the non-archimedean Lebesgue Dominated Convergence Theorem hold for the space $M(X,\mathbb{K})$? Only in the trivial case! We show (Theorem 2) that for every sequence $(f_{n})_n$ in $C(X,\mathbb{K})$ such that $f_{n}(x)\rightarrow 0$ for all $x\in X$ and $\| f_n \| \leq 1$ for all $n\in\mathbb{N}$, one has $\int_{X}f_{n}d\mu\rightarrow 0$ for each $\mu\in M(X, \mathbb{K})$ iff $X$ is finite. In the second part we characterize weakly Lindelöf non-archimedean Banach spaces $E$ with a base as well as Corson $\sigma(E',E)$-compact unit balls in their duals $E'$. We also look at the Kunen space from the non-archimedean point of view.