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I am studying Folland's text on measure theory and I am currently on section 2.4. In the beginning of this section he states

If $\{f_n\}$ is a sequence of complex-valued functions on a set $X$, the statement "$f_n \rightarrow f$ as $n \rightarrow \infty$" can be taken in many different sense, for example pointwise or uniform convergence. If $X$ is a measure space, one can also speak of a.e. convergence or convergence in $L^1$.

Before moving further with the text I wanted to clarify what he means by $f_n \rightarrow f$ a.e. I am familiar with the notion of almost-everywhere, but what kind of convergence is he referring to exactly? Is this pointwise convergence almost everywhere, uniform convergence almost everywhere? Something else entirely? It is possible he spoke about this earlier in the text but I could not find it.

This passage is before he introduces convergence in measure, so it cannot be that. In fact, he has a corollary that if $f_n \rightarrow f$ in $L^1$, there is a subsequence $\{f_{n_j}\}$ such that $f_{n_j} \rightarrow f$ a.e., and follows this with a remark saying

If $f_n \rightarrow f$ a.e. it does not follow that $f_n \rightarrow f$ in measure.

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  • $\begingroup$ Pointwise convergence a.e., as you said. $\endgroup$
    – lcv
    Mar 13 at 20:09

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This means pointwise convergence a.e. So the set of points in the space, for which $f_n$ does not pointwise converge to $f$ has measure $0$.

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