# Thread: Two Lebesgue Integration Problems

1. ## Two Lebesgue Integration Problems

1. Let f be a nonnegative measurable function on $\Re$. Show that if $F : \Re \rightarrow \Re$ is defined by $\int_{(0,x]} f(t)dm(t)$ then F is continuous on $\Re$.

I know that to prove that a function is continuous, you have to show that for any open set $A \subseteq \Re$ so that $f^{-1}(A)$ is open, but I'm not sure what to do from there.

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

I am clueless how to even start this problem.

2. For 1. use the sequential characterization of continuity. Take a sequence {x_n} converging to x. It suffices to show F(x_n) --> F(x).
A hint to do this is bring (0,x_n] in as a characteristic function and use an appropriate convergence theorem.

2. Let f be a nonnegative measurable function on a measurable set E with $\int_E f(x)dm(x)<\infty$. Prove that for each $\epsilon>0$ there exists $\delta>0$ such that for every measurable set $A \subseteq E$ with $m(A) < \delta$ we have $\int_A f(x)dm(x) < \epsilon$
For 2.: Let $\varepsilon>0$. Using the bounded convergence theorem, prove that there exists $M$ such that $\int_{\{f>M\}} f(x)dm(x)<\varepsilon/2$ (integration on the set where $f(x)>M$). Then choose $\delta<\varepsilon/(2M)$, and conclude using:
$\int_A f(x)dm(x)= \int_{A\cap\{f>M\}} f(x)dm(x)+\int_{A\cap\{f\leq M\}} f(x)dm(x)\leq \varepsilon/2 + M m(A)$.