I am familiar with the proof of showing that $f:[0,1]\to\mathbb{R}$, where $$f(x)=\begin{cases} 0 & \text{if} \, \, x\in\mathbb{R}\setminus\mathbb{Q} \\ 1 & \text{if} \, \, x \in \mathbb{Q} \end{cases}$$ is not Riemann integrable with the traditional definition of Riemann integration with Darboux sums and partitions of intervals, etc. However, I want to show the same result with an alternative definition. Additionally, I want to show that it is Lesbesgue integral using a particular definition as well.
Definition: A function $f$ is said to be Riemann integrable if for all $\epsilon>0$ there exists simple step functions $g,h$ such that $g \leq f \leq h$ and $\int h - g < \epsilon.$
Definition: By a simple step function we mean a finite linear combination of characteristic functions on semi-open intervals, i.e. if $s$ is a simple function then: $$s:=\lambda_1 \chi_{I_{1}} + \dots + \lambda_n \chi_{I_{n}}$$ where $\lambda_j$ are scalars and $I_j$ are semi-open intervals for $1 \leq j \leq n$.
Definition: A real valued function $f$ defined on $\mathbb{R}^n$ is said to be Lesbesgue integrable if there exists a sequence of simple step functions, $(f_n)$ such that the following two conditions are satisfied:
a) $\quad \displaystyle\sum_{n=1}^{\infty} \int |f_n| < \infty$
b) $\quad f(x)=\displaystyle\sum_{n=1}^\infty f_n(x) \quad \forall x \in\mathbb{R}^n$ such that $\displaystyle\sum_{n=1}^{\infty} |f_n| < \infty.$
Then, the integral of $f$ is defined to be: $$\int f = \sum_{n=1}^{\infty} \int f_n.$$
I'm not sure how to show the result with this definition, but I'm thinking of going about it by contradiction, suppose $f$ is Riemann integrable, then there are two simple step functions $g$ and $h$ such that $g \leq f \leq h$. Then, maybe we could choose a sequence $(g_n)$ and $(h_n)$ such that $g_n \to g$ and take subsequences such that each $g_n \in \mathbb{Q}$ for all $n$, and similarly $h_n \to h$ and take subsequences such that each $h_n \in \mathbb{R}\setminus\mathbb{Q}$, for all $n$, so $\int h - g = 1$, which is not less than $\epsilon$ for all $\epsilon>0$. This doesn't quite feel right though.
Then, as far as showing that it is Lesbesgue integable with this definition, I am thinking about enumerating a sequence of rational numbers $\left\{ q_1 , q_2 , \dots \right\}$ such that $f_n = q_n$ and using that but as my simple functions, but I'm having trouble formalizing it with this definition.