Hi: I need assistance with the following problem:
Let $(X,M,\mu)$ be a measure space. Let $X$ be the union of a countable ascending sequence of measurable sets $\{X_n\}$ and $f$ a nonnegative measurable function on $X$. Show that $f$ is integrable over $X$ if and only if there is an $N \geqslant 0$ for which $\int_{X_n} f~\text{d}\mu \leqslant N$ for all $n$.
I think I have to use these facts: Since $X_n\subset X_{n+1}$ and $X=\cup X_n$, $\mu(X) = \lim X_n$. For one direction, I know I have to show $\int_X fd\mu <\infty$.