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 can find a countable collection of open intervals $I_k$ such that
$$
E \subset U = \bigcup_{k \in \Bbb N} I_k, \qquad
\left(\sum_{k=1}^\infty m(I_k) \right) - m(E) < \epsilon
$$
Now, note that 
$$
\left| 
\int_U f\,dx - \int_E f\,dx 
\right| = 
\left|
\int_{U \setminus E} f\,dx
\right|
$$

Alternative: Show that $\int_U f\,dx = 0$ whenever $U$ is a $G_\delta$ set.  Then, note that there is a $U$ such that $E \subset U$ with $m(U \setminus E) = 0$.
