How would you interpret the following statement involving "a.e."?

Question

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.

Answer 1

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 $x \in X$.

Answer 2

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.