Lebesgue integration from definition

Question

I'd like to ask you a question about finding integral value using the Lebesgue definition. I've been trying to find a method to obtain a sequence of simple functions that are convergent to function under integral on some integral in $ \\R $. It`s quite easy to find a sequence of simple functions for functions $ x, x^2, x^3 $ (and any other polynomial). I was trying with other integrals $ \int_{0}^{1} sin(x) $, $ \int_{1}^{2} \frac{1}{x^3} $, $ \int_{1}^{4} \frac{1}{\sqrt{x}} $ but I can't find series forms for them. Do you know how to do it?

Answer 1

This is an interesting question that is seldom addressed in integration books. Here is an explicit formula for positive $f$, obtained by a dyadic decomposition of the y-axis. Note that the sum is finite if $f$ lies between two positive constants. $$ \int_X f d\mu = \lim_{r \rightarrow 1, r>1} \sum_{n\in {\bf Z}} r^n \mu\Bigl( f^{-1}([r^n, r^{n+1}))\Bigr). $$ This may be used to compute easily the integral of $x \mapsto x^\alpha$. The formula is discussed there.