Lebesgue Integral: Practice Problem

[nateHI]

Homework Statement

Suppose that f is a nonnegative Lebesgue measurable function and E is a measurable set.
Let A = {x ∈ E : f(x) = ∞}. Show that if $\int_E f dλ < ∞$ then $λ(A) = 0$

Homework Equations

The Attempt at a Solution

[/B]
Let $\phi(x)=\sum_{x\in A}a_i\mathcal{X}_{A_i}(x)$ so by the construction of $A$, $a_i=\infty$ for all $i$ and $A=\cup A_i$
Then
$\infty\ge\int_E f dλ= \int_{E-A}f dλ+\int_Af dλ=\int_{E-A}f dλ+\int \phi(x)=\int_{E-A}f dλ+\sum a_i \lambda(A_i)$
$\implies \lambda(A_i)=0$ for all i
$\implies \lambda(A)=0$
This seems simple and intuitive so I'm worried I'm missing something.

 

[HallsofIvy]

No, that's pretty much the idea.