Exercise 5, § 5

Let $(f_{mn})$ be a double sequence of measurable mappings of X into a metrizable space F. Assume that for every m, the sequence $(f_{mn})_{n \geq 1}$ converges locally almost everywhere to a function $g_m$, and that the sequence $(g_m)$ converges locally almost everywhere to a function h. Show that for every compact subset K of X, there exist two strictly increasing sequences $(m_k), (n_k)$ of integers $> 0$, such that the sequence of functions $f_{m_k, n_k}$ converges to h almost everywhere in K. (Note that for every $\varepsilon > 0$, there exists a compact set $K_1 \subset K$ such that $\mu(K - K_1) \leq \varepsilon$ and such that in $K_1$ the sequence $(g_m)$ and each of the sequences $(f_{mn})_{n \geq 1}$ are uniformly convergent.)