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Math Help - Sequences of Step Functions

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    Sequences of Step Functions

    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 $\displaystyle \lim_{m \rightarrow \infty} \int_{I} s_{m} \leq \int^{a + \delta + 3M}_{a + \delta} s_{N}$.
    Last edited by ragnar; July 21st 2010 at 10:15 PM. Reason: No longer have a question, really.
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