How would you interpret the following statement involving "a.e."? Here is an edited fragment from an exercise:


  Let $(X, \mathcal A, \mu)$ be a measure space, $(f_n)$ be a sequence of such and such functions. If $f(x)= \lim f_n(x)$ exists for almost every $x\in X$ then $f$ has such and such properties (in particular $f$ is measurable).


I'm confused about the role of $f$. Is it already given in the statement? E.g. if in the proof, one would set $f$ to zero on some convenient set of measure zero, it wouldn't be OK? (A proof which I encountered does exactly that.)

The question probably has little to do with measure theory but that a general "if something shows up in the assumption, must it be fixed from there on?" but I've added the context just in case.
If the measure space is not complete, then following statement is, in general, not correct.


  Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of measurable functions. If $f(x) = \lim_{n \to \infty} f_n(x)$ exists for almost every $x \in X$, then $f$ is measurable.


Just pick a non-measurable set $N$ such that there exists $M \supseteq N$ measurable with $\mu(M)=0$ (such a set $N$ exists if the measure space is not complete). Then $f_n(x) := 0$ converges for almost every $x \in X$ to $f(x) := 1_N(x)$, but $f$ is not measurable.

However, the following statement holds true:


  Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of measurable functions. If $f(x) = \lim_{n \to \infty} f_n(x)$ exists for almost every $x \in X$, then $f$ has a measurable modification, i.e. there exists a measurable function $\tilde{f}$ such that the set $\{x; \tilde{f}(x) \neq f(x)\}$ is contained in a $\mu$-null set.


Note that the function $\tilde{f}$ satisfies in particular $$\tilde{f}(x) = \lim_{n \to \infty} f_n(x)$$ for $\mu$-almost all $x \in X$.
Changing $f$ on a set of measure zero does not affect almost convergence to it. The converging sequence $f_n$ may or may not converge for those newly assigned values but it does not really care, as long as it maintains convergence on the complement of a measure-zero set.