I have a question about ‘almost-everywhere’.
I learned that in lebesgue-measure, we consider all (f=0 a.e) functions as a same function. In this view, we can choose a function for each coset (for same functions almost everywhere).
I know we can choose one function for each coset, but if we really need to choose the exact function, what would be the best definition? (I know the word ‘best’ is not a good term, but I don’t have any idea to explain it in mathematical way.)
For example, we should choose f=0 for {f=0 a.e}. And if we can choose continuous function, we should choose it. And if it there is a piecewise continuous function, we should choose the one that is (1) left-continuous or (2) right-continuous. (Anything would be okay, but anyway we should choose one standard.) And for a general coset... what(or how?) should i choose?
I want to construct this like we construct the ‘measure’ starting from open interval, then open set, than measurable set.
I guess there must be mathematical theory for this, but I don’t even know how to search... so please help.
Thank you so much, and sorry for my bad english.
I’ll really appreciate for any comments related to this. Thank you.