Prove that if a sequence \(f_1, f_2, \ldots\) of functions in \(L^1(\mathbb{R})\) converges to \(f\) in \(L^1(\mathbb{R})\), then there exists a subsequence \(f_{n_1}, f_{n_2}, \ldots\) that converges a.e. to \(f\).
Solution
1. Choose a subsequence with summable \(L^1\) errors.
Since \[ \|f_n-f\|_1\longrightarrow0, \] we can choose \[ n_1<n_2<\cdots \] so that \[ \|f_{n_k}-f\|_1<2^{-k} \] for every \(k\ge1\).
2. Apply Tonelli to the sum of the pointwise errors.
Because the summands are nonnegative, Tonelli’s theorem gives \[ \begin{aligned} \int_{\mathbb R}\sum_{k=1}^\infty |f_{n_k}(x)-f(x)|\,dx &=\sum_{k=1}^\infty\|f_{n_k}-f\|_1\\ &<\sum_{k=1}^\infty2^{-k}<\infty. \end{aligned} \] Therefore \[ \sum_{k=1}^\infty|f_{n_k}(x)-f(x)|<\infty \] for almost every \(x\in\mathbb R\). In particular, for almost every \(x\), \[ |f_{n_k}(x)-f(x)|\longrightarrow0. \] Hence \[ \boxed{f_{n_k}(x)\to f(x)\text{ for almost every }x\in\mathbb R.} \]