Sequences of Step Functions
Suppose a sequence of increasing step functions which approach a function on an unbounded interval , and a.e. on . I am asked to show that diverges, where this is the Lebesgue integral.
Here's what I have so far:
[Edit: Nevermind, I think I have it. I'll post what I did below, though, in case anybody is a) curious or b) sees a problem with my solution, since doesn't go into a whole lot of detail.]
Suppose converges to M. We may assume w.l.o.g. that is of the form , since we may extend the following argument to the other possibilities: and .
First, we select any and consider the interval . I claim that there is some $N$ such that for all . For suppose this is false, then there is some point at which for all , contradicting our assumptions. We then know that , which contradicts the fact that .