I was given a problem, which was to show that two functions are equal almost everywhere, where one function is measurable and the other is not.
My problem is that the set where the equality fails is not measurable, although it is a subset of a set of measure zero (the measure space is not complete). Thus, I feel like I'm trying to prove a contradiction, because by definition, the set where the equality fails should be of measure zero. How do you show this?
I've looked around, and I read somewhere that the set where the equality fails doesn't have to be of measure zero, as long as it's a subset of a set of measure zero, but no proof was given. I'm assuming it was a modified version of the definition I was given. Intuitively, it makes sense ... but is there an actual proof for this?
P.S. In case you want more details, the measurable function was the constant function 0, and the non-measurable function was the characteristic/indicator function (where the value is 1 on the non-measurable set).