Real Valued Lebesgue Intergal

Hello!

I'm trying to show that if f is a measurable function on some space X, and if

$\int{fd\mu} < \infty$

then $\mu\{x \in X : f(x) = \infty\} = 0$

In other words, trying to show f is real-valued (not infinity) mu-almost everywhere.

I thought maybe the contrapostive would work, ie,
Presume $\mu\{x \in X : f(x) = \infty\} > 0$ .
Call the set $\{x \in X : f(x) = \infty\} = B$ (for "big" valued, hehe)
Then $\int_B{fd\mu} = \int{f\chi_B d \mu}$
If I can just show $\int{f\chi_B d \mu} = \infty$ , then since $B \subset X$ ,
$\int_X{f d \mu} = \int{f d \mu} = \infty$ as well.

So I am just wondering if the above looks alright, and what's left is how to possibly show that $\int{f\chi_B d \mu} = \infty$ for $\{x \in X : f(x) = \infty\} = B$ and $\mu(B) > 0$ ?!
Thanks very much!