a function defined only upto a set of measure zero

Question

What does it mean that a function in Sobolev spaces (for example) to be defined only upto a set of measure zero?

Answer 1

There's nothing particular about Sobolev spaces here. If $(X,\mathcal{A},\mu)$ is a measure space and $Y$ is any set, we can define an equivalence relation on the set $\mathcal{F}(X,Y)$ of all functions from $X$ to $Y$ by saying that $f\sim g$ if $$\mu\big(\{x\in X \mid f(x)\neq g(x)\} \big) = 0.$$Check that this is an equivalence relation. The very common abuse of notation is to denote the equivalence class $[f]$ of $f$ by $f$ itself. Saying that $f$ is defined up to a set of measure zero means, effectively, specifying the equivalence class but not actually choosing a representative for such class.