Sequences of Step Functions

**ragnar** · July 21st 2010, 08:54 PM

Suppose $\{ s_{n} \}$ a sequence of increasing step functions which approach a function $f$ on an unbounded interval $I$ , and $f(x) \geq 1$ a.e. on $I$ . I am asked to show that $\{ \int s_{n} \}$ 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 $\{ \int_{I} s_{n} \}$ converges to M. We may assume w.l.o.g. that $I$ is of the form $[a, \infty)$ , since we may extend the following argument to the other possibilities: $(a, \infty), (-\infty, a], (-\infty, a),$ and $(-\infty, \infty)$ .

First, we select any $\delta > 0$ and consider the interval $$I_{0} = [a + \delta, a + \delta + 3M$]$ . I claim that there is some $N$ such that for all $$x \in I_{0}$,$ $n \geq N \Rightarrow s_{n}(x) \geq \frac{1}{2}$ . For suppose this is false, then there is some point at which $s_{N}(x) < \frac{1}{2}$ for all $N$ , contradicting our assumptions. We then know that $\displaystyle \frac{1}{2} \cdot 3M \leq \int^{a + \delta + 3M}_{a + \delta} s_{N}$ , which contradicts the fact that $\displaystyle \lim_{m \rightarrow \infty} \int_{I} s_{m} \leq \int^{a + \delta + 3M}_{a + \delta} s_{N}$ .

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