Q: Lebesgue Integral of ${x^2}$ over $[0,1]$ I know its just simple using Riemann Integral. I do not know how can I show the integral of ${x^2}$ over $[0,1]$ equals to $\frac13$ using definition of Lebesgue integral. 

A: In the case of continuous functions, it is known that the Riemann integral and the Lebesgue integral over a finite interval are equal. However, if you don't want to use this result, then what you can do is approximate $f(x) = x^2$ on $[0,1]$ from below by an increasing sequence of simple functions $f_n(x)$, and then calculating the integral of those simple function is just summing the areas of the corresponding rectangles. If you choose $f_n(x)$ is a systematic manner, then it should be possible to evaluate the limit $\lim_{n \rightarrow \infty} \int f_n(x)$, which equals $\int f(x)$ by the monotone convergence theorem.
