Exercise 3, § 1

a) Let $(f_\alpha)_{\alpha \in A}$ be a family of numerical functions $\geqslant 0$, directed for the relation $\leqslant$, such that the mapping $t \mapsto (f_\alpha(t))$ of $T$ into $\mathbf{R}^A$ is $\mu$-measurable. Let $f$ be the upper envelope of the family $(f_\alpha)$. Then $f$ is $\mu$-measurable and

(1)

b) Let $(f_\alpha)_{\alpha \in A}$ be a family of numerical functions $\geqslant 0$ defined on $T$ and having the following property: for every compact subset $K$ of $T$ and every $\varepsilon > 0$, there exists a compact set $K' \subset K$ such that $\mu(K - K') \leqslant \varepsilon$ and such that the restrictions to $K'$ of all the functions $f_\alpha$ are lower semi-continuous. Let $f$ be the upper envelope of the family $(f_\alpha)$. Show that $f$ is measurable and satisfies the relation (1).

Give an example of a mapping $t \mapsto (f_\alpha(t))$, satisfying the above condition, that is not $\mu$-measurable.