I am reading "Analysis on Manifolds" by James R. Munkres.
There is the following theorem in this book:
Theorem 11.3(p.96):
Let $Q$ be a rectangle in $\mathbb{R}^n$; let $f : Q \to \mathbb{R}$; assume $f$ is integrable over $Q$.
(a) If $f$ vanishes except on a set of measure zero, then $\int_Q f = 0$.
(b) If $f$ is non-negative and if $\int_Q f = 0$, then $f$ vanishes except on a set of measure zero.
I think we don't need to assume that $f$ is integrable over $Q$.
Since $f$ vanishes except on a set of measure zero, $f$ is a bounded continuous function except on a set of measure zero. So, $f$ is integrable over $Q$.
So, I think we don't need to assume that $f$ is integrable over $Q$ for (a).
When we write $\int_Q f$, we already assume that $f$ is integrable over $Q$.
So, I think we don't need to assume that $f$ is integrable over $Q$ for (b).
Am I right or not?
Best Answer
You are right when you state that if $f$ vanishes except on a set of measure zero, then $f$ is integrable. But if you don't say that $f$ is integrable, then it doesn't make sense to talk about $\int_Qf$. Yes, if $f$ wasn't integrable, this wouldn't make sense, but it is not a good idea to leave those assumptions implicit.