Exercise 3, § 1

In the example of Exer. 2, take for Y the discrete space $\mathbf{N}$. For every $n \in \mathbf{N}$, let $e_n$ be the function in $\mathscr{K}(X; \mathbf{R})$ such that $e_n(n) = 1$ and $e_n(x) = 0$ for $x \neq n$ in X. Show that the set A formed by the functions 0 and $e_n/(n+1)$ is compact in $\mathscr{K}(X; \mathbf{R})$, but not strictly compact. If $\mathfrak{U}$ is the ultrafilter on $\mathbf{N}$ induced by the filter of neighborhoods of $\omega$ in $\beta Y = \beta \mathbf{N}$, show that, with respect to $\mathfrak{U}$, the family of positive measures $(n \varepsilon_n)$ tends to 0 uniformly in every strictly compact subset of $\mathscr{K}(X; \mathbf{R})$, but does not tend uniformly to 0 in A.