$h \in L^{1} (\Omega , m)$ such that $hm=0$ then show that $h=0$.

Question

The following question was asked in my assignment of measure theory and I am not able to make much progress on it.

Question: Let $(\omega , B)$ be a measurable space and let m be a positive measure on $(\Omega , B, m)$ . Show that if $h \in L^{1} ( \Omega ,m)$ is such that hm=0 , then h=0.

I am not sure how exactly to use the given condition $hm=0$ .I have been following the book mathematical analysis by Tom Apostol.

Can you please help me with this problem?

Answer 1

First, let's clarify what $hm$ means. In your context, $hm$ is a measure on the space $(\Omega,B)$ defined as follows: $$hm(E) = \int_E h\text{d}m,\quad E\in B,$$ So what you have to prove is that if $\int_E h\text{d}m=0$ for all $E\in B$, then $h=0$ (a.e.). Here is a reference for that.