Let us consider the following Dirichlet function $D: [0,1] \subset \mathbb{R} \to \mathbb{R}$: $$ D(x) = \begin{cases} 1, \quad x\in\mathbb{Q} \\ 0,\quad \mathrm{else}.\end{cases}$$
I would like to show that it is Lebesgue integrable and compute the value of the integral if I am defining it through the completion of the space of all continuous compactly supported Riemann integrable function $C_c(\mathbb{R})$, call this completion $L^1(\mathbb{R})$.
As far as I understand, consider the space $C_c(\mathbb{R})$ and define its metric to be the following. If $f$ is nonzero on an interval $I_1$ and $g$ is nonzero on an interval $I_2$ then $I = I_1 \cup I_2$ is also an interval, so Riemann integral can be defined. $$ d(f,g) = \int \limits_{I = I_1 \cup I_2} |f - g|$$
Then, define completion with respect to this metric. Before computing the value of the integral, first I have trouble understanding how to show that $D \in L^1(\mathbb{R})$. Each element $X \in L^1(\mathbb{R}$) is an equivalence class $[f]$ of Cauchy sequences of $\{ f_k \}_{k \in \mathbb{N}}$ such that equivalence is defined for two Cauchy sequences $\{ f_k \}_{k \in \mathbb{N}}$, $\{ g_k \}_{k \in \mathbb{N}}$ if and only if: $$ \lim \limits_{k \to \infty} d(f_k, g_k) = 0 $$
I understand that if $f \in C_c(\mathbb{R})$ then we can identify it with an element $X \in L^1(\mathbb{R})$ which is equivalence class $[f]$ in which the Cauchy sequence defined by $f_k = f$ belongs to. And, intuitively, we would like to say that $D$ might be approximated by compactly supported continuous functions, but I don't see how this fits in this framework. Honestly, I don't even see how to identify arbitrary $X \in L^1(\mathbb{R})$ with a function in general. I guess also that from the point of view of applications, I could use identification using pointwise convergence, but I am not sure if this is what is intended.
Question 1: How to identify $D$ with such classes of Cauchy sequences from $C_c(\mathbb{R})$? Can you give me both general algorithm and concrete example of a Cauchy sequence for this function $D$?
Let us assume it is understood how to identify $D \in L^1(\mathbb{R})$ with a class of Cauchy sequences. In that case, there is supposed to be a Cauchy sequence $\{ f_k \}_{k \in \mathbb{N}}$, where for each $k$, we can define integral.
Question 2: Is it correct to say that the Lebesgue integral of $D$ is then defined to be the following?
$$ \int D = \lim \limits_{k \to \infty} \int_{I_k} f_k $$
Thanks a lot, and I would really appreciate some beginner references related to this question.