How Lebesgue integration solved the problem of a function being integrable but its limit is not integrable?

Question

My professor gave us the following form of Dirichlet function as an example of the problems we faced in Riemann integration:

$\{r_{n}\}$ enumeration $\mathbb{Q} \cap [0,1]$

$$ f_{n}(x) = \begin{cases} 1 & \quad x \in \{r_{1}, ... , r_{n}\} \\ 0 & \quad \text{otherwise}. \end{cases} $$

And he said that: each $f_{n}$ is integrable but its limit is not integrable.

My questions are:

1- why each $f_{n}$ is integrable but its limit is not integrable?

2- How did Lebesgue integration solve this problem?

Could anyone help me understand answers to these questions, please?

Answer 1

1. Each $f_n$ is integrable because you can check that each lower sum is equal to $0$ and that the upper sums can take (positive) values as small as you want. Therefore, the (Riemann) integral of $f$ is $0$. However, if $f$ is the limit, then every lower sum is $0$ and every upper sum is $1$. Therefore, $f$ is not Riemann-integrable.
2. In the case of Lebesgue integration, $f$ is integrable, and its integral is $0$.