Exercise 6, § 3

a) Show that in all of the results of §3 in which the concept of a vaguely continuous mapping $t \mapsto \lambda_t$ intervenes, one can replace this hypothesis by the following: for every function $g \in \mathscr{K}_+(X)$, the function $t \mapsto \lambda_t(g)$ is lower semi-continuous on T.

b) Consider a scalarly essentially integrable mapping $t \mapsto \lambda_t$ of T into $\mathscr{M}_+(X)$ that satisfies the following condition: For every compact subset K of T and every $\varepsilon > 0$, there exists a compact set $K_1 \subset K$ such that $\mu(K - K_1) \leq \varepsilon$ and such that the restrictions to $K_1$ of all the functions $t \mapsto \langle f, \lambda_t \rangle$, where $f$ runs over $\mathscr{K}_+(X)$, are lower semi-continuous. Show that the mapping $t \mapsto \lambda_t$ is $\mu$-adequate.