I am trying to understand the term "almost everywhere" from measure theory correctly. So given two extended real-valued integrable functions $f, g: X \rightarrow \bar{\mathbb{R}}$ with $$\int_A f d\mu = \int_A g d\mu, $$ for all measurable $A$, then it follows that $f=g$ almost everywhere, i.e., that $f$ and $g$ are equal except on a $\mu$ nullset. What I now dont understand is, if there is one specific null set on which they dont agree or if they dont agree on all null sets.
But if they dont agree on all null sets this would be weird, since we could (with $\mu$ the Lebesgue measure), for example, always take a countable number of real numbers, which form a nullset, and $f$ and $g$ wouldnt be allowed to agree on this set. But then we could take another such set, again with measure 0, and again $f$ and $g$ wouldnt be allowed to agree on them. And we could go on and on like this and in the end there wouldnt be any set left on which $f$ and $g$ would actually still be equal...
"Almost everywhere" means that the set where whatever-it-is fails is a null set. If you have two things happening "almost everywhere" then the sets where they fail will in general be different null sets. (But it's OK because e.g. the set where at least one of them fails is the union of those two null sets, and the union or two null sets is a null set.)
If something holds outside every null set then it holds literally-everywhere because the empty set is a null set.
If something fails-to-hold on every null set then it holds literally-nowhere because every singleton set is a null set.