第 4 章 随堂习题#
设 \(\{f_n(x)\}\) 是可测集 \(E\) 上的非负可测函数列, 证明对任意 \(n \in \mathbb{N}\), 有
\[\int_E \left( \inf_{k \geqslant n} f_k(x) \right) ~ \mathrm{d} m \leqslant \inf_{k \geqslant n} \int_E f_k(x) ~ \mathrm{d} m.\]
设 \(\varphi = \chi_{(\alpha, \beta)}\), \((\alpha, \beta) \subset [-\pi, \pi]\), 其 Fourier 级数为
(1)#\[\frac{A_0}{2} + \sum_{n=1}^{\infty} \left( A_n \cos nx + B_n \sin nx \right),\]其中
\[\begin{split}A_n & = \frac{1}{\pi} \int_{-\pi}^{\pi} \varphi(x) \cos nx ~ \mathrm{d} x = \frac{1}{\pi} \int_{\alpha}^{\beta} \cos nx ~ \mathrm{d} x = \frac{1}{n \pi} \left( \sin n\beta - \sin n\alpha \right), \\ B_n & = \frac{1}{\pi} \int_{-\pi}^{\pi} \varphi(x) \sin nx ~ \mathrm{d} x = \frac{1}{\pi} \int_{\alpha}^{\beta} \sin nx ~ \mathrm{d} x = \frac{1}{n \pi} \left( \cos n\alpha - \cos n\beta \right).\end{split}\]考虑级数 (1) 的前 \(n\) 项部分和
\[\varphi_n(x) := \frac{A_0}{2} + \sum_{k=1}^{n} \left( A_k \cos kx + B_k \sin kx \right),\]证明 \(\varphi_n(x)\) 可由 Dirichlet 核 \(D_n(u) = \frac{\sin \left( n + \frac{1}{2} \right) u}{\sin \frac{u}{2}}\) 表示为
\[\varphi_n(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \varphi(x+u) D_n(u) ~ \mathrm{d} u = \frac{1}{2\pi} \int_{-\pi}^{\pi} \varphi(x+u) \dfrac{\sin \left( n + \frac{1}{2} \right) u}{\sin \frac{u}{2}} ~ \mathrm{d} u.\]
设 \((X, \mathscr{R}), (Y, \mathscr{S})\) 是两个可测空间, \(E \subset X \times Y\) 是 \(X \times Y\) 的一个可测集, \(f : E \to \mathbb{R}\) 是 \(E\) 上的可测函数. 证明 \(f\) 的每个截口都是可测函数.