What does $f = g$ almost everywhere mean?

Question

I need to prove the following proposition.

If $f,g: [a,b] \rightarrow \mathbb{R}, $ $f$ is Lebesgue measurable, and $ f = g $ almost everywhere, then $g$ is Lebesgue measurable. But I do not know what exactly is almost everywhere.

Answer 1

This terminology is from measure theory. Two functions are said to be equal almost everywhere on $[a,b]$ if there exists a set $E\subseteq [a,b]$ of measure $0$ such that $f(x)=g(x)$ for all $x\in [a,b]\setminus E$.

A subset $E\subseteq [a,b]$ is said to have measure $0$ if for every $\epsilon>0$ there exist countably many intervals $((a_k, b_k))_k$ such that $E\subseteq \bigcup_k (a_k, b_k)$ and $\sum_k (b_k-a_k) < \epsilon$.

How you can go about showing that if $f$ is measurable then so is $g$ depends on the definition you are using. Either way, I hope this helps!