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.