How do I compute Lebesgue integrals of non-simple functions? I know this is not the purpose of Lebesgue integration, but it would still be nice to know how to do this. For example, how do I compute the integral
$$\int_{[0,1]} \sin(x)\,\mu(\mathrm dx).$$
I know how to compute it the Riemann way—I just want to get practice on Lebesgue integration.
I know I have to make $\sin x$ a pointwise limit of simple functions which would be $$\lim_{n \rightarrow \infty} \frac{\left \lfloor{10^n\cdot\sin x}\right \rfloor}{10^n}.$$
 A: I take it that $\mu$ here is the Lebesgue measure on $\mathbb{R}$. Assuming such, you actually calculate your integral exactly how you'd calculate a Riemann integral and there equal. Reason being, what you have here is the Lebesgue integral of a bounded measurable function over a set of finite measure. As such, there is the following theorem,

Theorem: Let $f$ be a bounded function defined on the closed, bounded interval $[a, b]$. If $f$ is Riemann integrable over $[a, b]$, then it is Lebesgue integrable over $[a, b]$ and the two integrals are equal.

A: Since the sequence of simple functions you gave converges uniformly, you can simply evaluate the integrals of the simple functions, and then take the limit of that sequence.
$$\int_{[0,1]} \sin(x)\,\mathrm{d}(\mu x) = \lim_{n \rightarrow \infty} \int_{[0,1]} \frac{\left \lfloor{10^n\cdot\sin(x)}\right \rfloor}{10^n}\,\mathrm{d}(\mu x)$$
Since $\sin(x)$ is monotone on $[0,1]$, the integral of the simple functions can be expressed by the following sum:
$$\sum_{k=0}^{10^n-1} 10^{-n}k\, \mu\!\left([\arcsin\lfloor10^{-n}k\rfloor,\arcsin\lfloor10^{-n}(k+1)\rfloor)\right) + \mu(\{1\})$$
