Here is what I'm considering problem. Please give a more thougt
Let $f$ be an integrable function on a measure spaxe $(X,M,\mu)$ such that
$$ \int_{E}fd\mu=0$$
for all sets $E \in M$. Prove that $f=0$ $\mu$-a.e.
I can solve this problem If we assume $f$ is non-negative.
So I try to show $\int_{E}fd\mu=0$ iff $\int_{E}|f|d\mu=0$
But I cannot prove it. I guess there is counterexample ($\int_{E}|f|d\mu$ is non zero but $\int_{E}fd\mu=0$).
Please give me any other way! Thank you
On
Of course you cannot prove $\int_E f d\mu = 0$ iff $\int_E |f| d\mu = 0$. You need to add the assumption "$\forall E$". If $E$ is fixed a simple counterexample is the sine function.
However, split $f = f^+ - f^-$ where $f^+(x), f^-(x) \geq 0$. Then $|f|= f^+ + f^-$.
Now supp($f^+$) $\cap$ supp($f^-$) = $\emptyset$ ... Can you solve the problem now?
Try integrating over $$E_+:=\{x\in M: f(x)\geq 0\}$$ and $$E_-:=\{x\in M: f(x)\leq 0\}.$$ Then use what you already have about nonnegativ functions.