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It is well known that if a sequence of $L^2$ functions $(f_n)_{n \in \mathbb{N}}$ converges to a function $f$, i.e. $\lim_{n\in\mathbb{N}} \int \lVert f_n - f \rVert^2 dx = 0$, then there exists a subsequence $(f_{n_k})_{k\in \mathbb{N}}$ such that $f_{n_k}$ converges to $f$ a.e.. The question is under which (sufficient) conditions the entire sequence converges a.e.. I am particularly interested in the case where the underlying space is a probability space, if that makes any difference.

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There is no reason to believe that $L^2$-convergence gives any information on $\mathbb{P}$-almost sure convergence. One usual counterexample is the 'running boxes', which can be also realized by continious functions. Another example is $f_n(x) = \sin(2\pi n)^{n}$ on $([0,1],\lambda)$, see here.

Assuming that $f_n = \sum_{k=1}^n X_n$ with independent $X_n$, then (by a theorem of Levy) we know that $f_n \rightarrow f$ a.e. is equivalent to stochastic convergence. Thus, in this case we would also get a.e. convergence.

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I'm not aware of any simple conditions under which convergence in $L^2$ implies convergence almost everywhere, other than things that are too trivial to be useful or interesting, like (i) if the measure space is purely atomic or (ii) if $||f_n-f||_2\to0$ and $f_n\to g$ almost everywhere then $f_n\to f$ almost everywhere.

A striking example: Say $f$ is a $2\pi$-periodic square integrable function on the line and let $s_n(f)$ be the $n$-th partial sum of the Fourier series. The fact that $||s_n(f)-f||_2\to0$ might or might not be regarded as trivial, but it's certainly no big deal, something everyone learns in a first year course on measure theory. On the other hand, Carleson's theorem, that $s_n(f)\to f$ almost everywhere, is still after all these years a hugely difficult theorem. Hugely difficult even assuming $f$ is continuous.

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Sorry for the fuzzy question and thank you all for the interesting answers which made me realize that the property I was look for is monotonicity, i.e. we assume $$ f_{n+1} \le f_n, \quad \forall n\in \mathbb{N} $$ and as mentioned in my question that $\lim_{n\in\mathbb{N}} \int \lVert f_n - f \rVert^2 dx = 0$, then we can deduce a.s. convergence.

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