An integrable Functions is almost everywhere finite
An integrable Functions is almost everywhere finite
Attempt: Let $X = \{ x : f(x) \, \text{ is infinite}\} $. We must show $m(X) = 0$. Suppose $m(X) > 0 $. then on $X$, we have
$$ \int\limits_X f \, dm > \infty$$
which implies $f$ cannot be integrable: contradiction.
Is this a correct solution? thanks
Look at $[|f| > n]$; you have $|f| \ge |f|\chi_{[|f| > n]}$ and apply dominated convergence.