The question:
The standard definition of singular measures is given on Wikipedia:
Two positive measures $\mu$ and $\nu$ defined on a measurable space $(\Omega,\Sigma)$ are called singular if there exist two disjoint sets $A,B\in\Sigma$ whose union is $\Omega$ such that $\mu$ is zero on all measurable subsets of $B$ while $\nu$ is zero on all measurable subsets of $A$.
I was wondering what if we weaken the definition to not require $A$ and $B$ to be measurable sets; the rest of the definition is unchanged. Is it then equivalent to the original definition?
"Motivation"
I was trying to do this because I misread the original definition and then I was thinking how nice would it be if $A$ and $B$ were measurable... Then I read the definition again, but the wondering is still here.
I tried some simple tricks like picking a set (call it $X$) at a time which is zero in one measure and non-zero in the other and then define $A$ and $B$ through transfinite recursion (unioning all the sets I picked so far), but I wasn't able to do that either. There is an equivalent definition of absolutely continuous measures, defined in $\delta-\varepsilon$ notation which seems to yield something like "I can pick such $X$ iff the two measures are not absolutely continuous one w.r.t. to the other". But I get lost here... It is as far as my intuition goes. However, I still doubt it could be done that way.
Is anything known about this weakened definition?
Note: The opposite question was asked here and the answer to this question given by Eric Wofsey answers exactly that other question.
This weakened definition is much weaker than the usual definition, to the point of being basically useless. For instance, it can be shown that there exists a subset $A\subset\mathbb{R}$ (e.g., a Bernstein set) such that both $A$ and its complement intersect every set of positive Lebesgue measure. That is, every Lebesgue measurable subset of either $A$ or its complement has measure $0$. This means that by your weakened definition, Lebesgue measure is singular with respect to itself!