Here is the problem statement:
For each of the following functions $f(x)$, find out whether its integral $\int_{0}^{1} f(x) dx$ exists as a Lebesgue integral with respect to the uniform measure on $[0,1]$, and whether it exists as an improper Riemann integral. $$\begin{aligned} &1. &f(x) = \frac{1}{\sqrt{x}} \\ &2. &f(x)= \left\{ \begin{aligned} &1, &x \text{ is rational} \\ &0, &x \text{ is irrational} \end{aligned} \right. \\ &3. & f(x) = \frac{\sin(1/x)}{x}\end{aligned}$$
I have no problem determining whether the improper Riemann integral exists. For example, it exists for the first function, does not exist for the second function, and exists for the third function.
However, I've just started learning Lebesgue integration, so this is giving me some trouble. In this particular class, we are using the approach to Lebesgue integration using indicator functions and elementary functions. In particular, if $(X, \Sigma, P)$ is a probability space and $I_{A}$ is the indicator function for some set $A \in \Sigma$, then $$\int I_{A} dP = P(A).$$ Next, if $f$ is an elementary function, meaning it takes countably many values $f_{i}$, then $$ \int f dP = \sum_{i=1}^{\infty} f_{i} P( \{ x \mid f(x) = f_i\}).$$ Finally, if $f$ is an arbitrary function and $f_{n}$ is a sequence of elementary functions that converges uniformly to $f$, then $$\int f dP = \lim_{n \to \infty} \int f_{n} dP.$$
So, my problem with determining whether these functions are Lebesgue integrable or not boils down to me being unable to determine whether or not there is a sequence of elementary functions that converges uniformly to the given function.
Any help would be highly appreciated.