I was studying the dominated convergence theorem and the question came up.
My question is as follows.
If $f \in {\cal L}^1(X,\bar{\cal M},\bar{\mu})$, is it true that $f \in {\cal L^1}(X,{\cal M},\mu)$ and $\int fd\mu = \int f d\bar{ \mu}$ ?
I guess it is not true because f isn’t nessesarily ${\cal M}$-measurable function.
But what if there is ${\cal M}$-measurable function g such that f=g almost everywhere?
Thank you.