If the sequence of Lebesgue measurable functions \(\{f_n\}_{n=1}^\infty\) on \(\mathbb{R}^d\) satisfying that
\[\lim_{n \to \infty} \int_{\mathbb{R}^d} |f_n(x)|^2 \, dx = 0,\]
show that there exists a subsequence of functions \(\{f_{n_j}\}_{j=1}^\infty\) such that
\[f_{n_j}(x) \to 0 \quad \text{a.e. } x.\]
Solution
Since \[ \|f_n\|_2^2=\int_{\mathbb R^d}|f_n(x)|^2\,dx\longrightarrow0, \] we may choose a subsequence \((f_{n_j})\) such that \[ \|f_{n_j}\|_2^2\le 2^{-j} \qquad(j\ge1). \] Therefore \[ \sum_{j=1}^\infty \int_{\mathbb R^d}|f_{n_j}(x)|^2\,dx \le \sum_{j=1}^\infty 2^{-j}<\infty. \] By Tonelli’s theorem, \[ \int_{\mathbb R^d}\sum_{j=1}^\infty |f_{n_j}(x)|^2\,dx =\sum_{j=1}^\infty\|f_{n_j}\|_2^2<\infty. \] Hence \[ \sum_{j=1}^\infty |f_{n_j}(x)|^2<\infty \] for almost every \(x\). For every such \(x\), the terms of this convergent nonnegative series tend to zero, so \[ |f_{n_j}(x)|^2\to0. \] Thus \[ \boxed{f_{n_j}(x)\to0\quad\text{for a.e. }x\in\mathbb R^d.} \]