TITLE: $f=\infty$ on a set of measure 0, then $\int_E f = 0$
QUESTION [7 upvotes]: Let $E$ be a set of measure zero and define $f = \infty$ on $E$. Show that $\int_E f = 0$.
This is out of Royden 4E, p 84.
I know how to prove this if $f=0$ on $E$.  But I'm curious, as stated, won't this result in a situation in which $\infty \cdot 0$.

REPLY [2 votes]: Another way would be to define $f_n=n$ on E and then apply the monotone convergence theorem. So $\int_{E}f=\lim_{n}\int_{E}f_n=\lim_{n}\int_{E}n=0$ , all the limits are as n goes to infinity.