Thread: Two Lebesgue Integration Problems

1. Let f be a nonnegative measurable function on . Show that if is defined by then F is continuous on .

I know that to prove that a function is continuous, you have to show that for any open set so that 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 . Prove that for each there exists such that for every measurable set with we have

I am clueless how to even start this problem.

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.

Originally Posted by grad444

2. Let f be a nonnegative measurable function on a measurable set E with . Prove that for each there exists such that for every measurable set with we have

I am clueless how to even start this problem.

For 2.: Let . Using the bounded convergence theorem, prove that there exists such that (integration on the set where ). Then choose , and conclude using:

.