# Under which conditions $L_1$ convergence implies pointwise a.e. convergence, in a finite measure space?

Consider a measure space $$(X, \mathcal{X}, \mu)$$, with $$\mu(X)< \infty$$. For measurable functions $$(f_n)_{n \geq 1}, f$$ we know that

$$\Vert f_n -f \Vert_{L_1(\mu)}:=\int_X|f_n(x)-f(d)|\mu(dx)\to_{n \to \infty}0$$

entails that there exists a subsequence $$(f_{n_j})$$ converging to $$f$$ almost uniformly and, hence, pointwise almost everywhere. Under which conditions such a statement can be extended to the entire sequence $$(f_n)_{n \geq 1}$$?

I know that a relatively strong additional condition that does the job is $$\sum_{n \geq 1}\Vert f_n -f \Vert_{L_1(\mu)} <\infty.$$

Is there anything milder, concerning for example continuity or nonnegativity of $$f_n$$ and $$f$$? In particular, what if, for example, $$X=[0,1]^d$$, for some $$d \in \mathbb{N}_+$$, and $$\mu$$ is the Lebesgue measure? In this case could we combine Theorem 5 in

https://terrytao.wordpress.com/2010/10/02/245a-notes-4-modes-of-convergence/

and possibly some relation between convergence in (Lebesgue) measure with pointwise a.e. convergence?

• The criterion is called Stein's maximal principle. Jul 23, 2019 at 8:45
• Actually, the operator formulation of Stein's maximal principle makes it a bit hard to digest to me. Would you mind giving some hints in a $(f_n)$ sequence formulation, as considered above? Jul 23, 2019 at 9:12
• This is an interesting question. The "fast L1 convergence" of Exercise 5 of the linked Tao's blog page gives almost uniform convergence, which is more than you want. I conjecture that the corresponding criterion for a.e. convergence is the one in my community wiki answer, but I have not proven that and I am not even sure it is correct. Jul 23, 2019 at 14:23
• Since in my question I'm only considering a finite measure space, by Egorov’s theorem a.e. pointwise convergence and almost uniform convergence are equivalent (e.g. Theorem 2 in terrytao.wordpress.com/2010/10/02/…). Outside a finite measure framework, of course, almost uniform convergence is more than I want. Jul 23, 2019 at 15:32

If $$\|f_n\|_{L^1}\to 0$$, the maximal principle of Stein makes me think that a necessary and sufficient condition for a.e.-pointwise convergence $$f_n(x)\to 0$$ is $$\sup_{t>0} t \left\lvert \{x\ :\ f^\star(x)>t\}\right\rvert <\infty,$$ where $$\lvert\cdot\rvert$$ denotes the measure of a set, and $$f^\star(x):=\sup_{n\in\mathbb N} |f_n(x)|.$$