Originally Posted by

**grad444** 2. Let f be a nonnegative measurable function on a measurable set E with $\displaystyle \int_E f(x)dm(x)<\infty$. Prove that for each $\displaystyle \epsilon>0$ there exists $\displaystyle \delta>0 $ such that for every measurable set $\displaystyle A \subseteq E$ with $\displaystyle m(A) < \delta$ we have $\displaystyle \int_A f(x)dm(x) < \epsilon$

I am clueless how to even start this problem.

For 2.: Let $\displaystyle \varepsilon>0$. Using the bounded convergence theorem, prove that there exists $\displaystyle M$ such that $\displaystyle \int_{\{f>M\}} f(x)dm(x)<\varepsilon/2$ (integration on the set where $\displaystyle f(x)>M$). Then choose $\displaystyle \delta<\varepsilon/(2M)$, and conclude using:

$\displaystyle \int_A f(x)dm(x)= \int_{A\cap\{f>M\}} f(x)dm(x)+\int_{A\cap\{f\leq M\}} f(x)dm(x)\leq \varepsilon/2 + M m(A)$.