Show that $\int _E f=0$ for each subset $E $ of $\mathbb R $ of finite Lebesgue measure Let $ f : \mathbb R \rightarrow \mathbb R$ be a bounded Lebesgue measurable function such that $\int_a^b f =0$ for all real $a,b.$ 
Show that $\int _E f=0$ for each subset $E $ of $\mathbb R $ of finite Lebesgue
measure
Actually I am new to measure theory.So maybe above is simple I can't proceed

Hint: We have, $$f = f^+-f^-$$ where $f^+$ and $f^-$ denote the positive and negative part of $f$, respectively. By assumption, the ($\sigma$-finite) measures
$$\nu(dx) := f^+(x) \, dx \qquad \mu(dx) := f^-(x) \, dx$$
satisfy
$$\mu((a,b)) = \nu((a,b)).$$
Conclude from the uniqueness of measure theorem that $\mu = \nu$ on $\mathcal{B}(\mathbb{R})$.
