Monotone Convergence Theorem Homework: Integrals & Increasing Sequences

[Ted123]

Homework Statement

Homework Equations

Monotone Convergence Theorem:

http://img696.imageshack.us/img696/5469/mct.png

The Attempt at a Solution

I know this almost follows from the theorem. But I first need to write $\displaystyle \int_{I_n} f = \int_S f_n$ for some $f_n$ in such a way that $(f_n)$ is an increasing sequence tending to $f$. (Then we have something that satisfies the hypotheses of the theorem.) What $f_n$ could I use?

Then in the case of any function $g$ can I consider positive and negative parts?

 

[Chaos2009]

Hmm, what if you let $f_{n} \left( x \right) = \left\{ \begin{array}{rl} f \left( x \right) &, x \in I_{n} \\ 0 &, x \not \in I_{n} \end{array} \right.$. I'm not sure if $f_{n} \in \mathcal{L}^{1} \left( \mathbb{R}^{k} \right)$ but it is an increasing sequence of functions which converges point-wise to $f$.