TITLE: If a function is positive on a set of measure greater than zero, is the Lebesgue integral of that function greater than zero?
QUESTION [7 upvotes]: Suppose we have a set $A \subset \mathbb{R}^n$ such that $f(x) > 0$ for $x \in A$ and $m(A) > 0$. 
Does it follow that $\int_A f > 0$?
Obviously if there is some kind of lower bound on $f(x)$ on some non-trivial subset of $A$, then we're done, but is it possible for there not to be any such lower bound? I'm thinking like $f(a)$ is some number and then the value of $f$ decreases very rapidly everywhere on $A$.

REPLY [14 votes]: Let $A_n = A \cap \{ f > \frac{1}{n} \}$.  Then $A = \cup A_n$.  If each $A_n$ had measure $0$, then $A$ itself would have measure $0$.  It follows that $m(A_n) > 0$ for some $n$.  So $\int_A f \geq \int_{A_n} f \geq \frac{1}{n} m(A_n) > 0$.