Exercise 2, § 3

Let B be a locally compact space, $(\nu_\iota)_{\iota \in I}$ the family of all positive measures on B. Let T be the product space I × B, I being equipped with the discrete topology, and let $\nu$ be the measure on T whose restriction to $\{\iota\} \times B$ is the canonical image of the measure $\nu_\iota$ on B, for all $\iota \in I$. Let p be the projection of T onto B; show that if there exists an uncountable compact subset of B, then $\nu$ does not admit a pseudo-image measure under p. (Show that every point of B would have measure > 0 for such a measure.)