Measurable Function and Sequence of Real Numbers

Let be a Lebesgue measurable subset of with and let be a sequence of real-valued Lebesgue measurable functions on . Prove or disprove that there exists a sequence of positive real numbers such that almost everywhere on .

I suppose the statement is true? So far can't think of any counterexample. If it's true, how do I go about starting the proof?

Thanks in advance.

Re: Measurable Function and Sequence of Real Numbers

Re: Measurable Function and Sequence of Real Numbers

Hey thanks very much! Anyway I would like to know why we need to establish S_k = {x \in E : |f_n(x)| > k}. Is it because (f_n) is a sequence of real-valued Lebesgue measurable functions without any other condition given?

Thanks in advance.

Re: Measurable Function and Sequence of Real Numbers

Quote:

Originally Posted by

**Markeur** Hey thanks very much! Anyway I would like to know why we need to establish S_k = {x \in E : |f_n(x)| > k}. Is it because (f_n) is a sequence of real-valued Lebesgue measurable functions without any other condition given?

If you knew that the functions were bounded then the result would be easy to prove. But a measurable function, even on a set of finite measure, need not be bounded. So the idea is to use the measurability condition to find a "large" subset of E on which is bounded. But the details of the proof are quite delicate, and I have had to edit my previous comment a couple of times to get an argument that (I think!) works.