习题 5, § 5

令 $(f_{mn})$ 为 X 的可测量映射的双序列转换为可测量的空间 F。 假设对于每 m，序列 $(f_{mn})_{n \geq 1}$ 几乎处处局部收敛到函数 $g_m$，并且序列 $(g_m)$ 几乎处处局部收敛到函数 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.)