I've a question about the definition of a property almost everywhere. On the German Wikipedia site, the definition is:
Let $(X,\mathcal{A},\mu)$ be a measured space.
A property $A$ holds almost everywhere in $X$ if and only if $\exists N\in \mathcal{A}$ with $\mu(N)=0$ and $A$ is true for all $x\in X\backslash N$.
The next sentences confuses me (I translate):"Note, the set where $A$ does not hold, need not to be measurable."
But I thought, we can deduce from the definition: The set where $A$ does not hold has measure $0$.
For example suppose we know that a measurable function is almost everywhere discontinuous. Now let $B:=\{x\in X; f \mbox{ is continuous in }x\}$. Is the measure of $B$ equal zero?
Thanks for your help
hulik
Best Answer
This is true if the measure space is complete, that is if every subset of a measurable set with measure zero is measurable too.
In general, it can be that you have a measurable set $N$ with $\mu(N)=0$ and a nonmeasurable set $M\subseteq N$. If a property holds exactly for the elements of $X\backslash M$, it also holds for every element in $X\backslash N$ and therefore almost everywhere, but the set of elements on which it holds, $X\backslash M$ is not measurable.