Formula for an integration on $\mathbb{Q} \cap [0,1]$

Question

In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting:

$$ \int\limits_{\mathbb{Q}\cap [0,1]} f(x) dx = \lim_{Q \to \infty} \frac{2}{(Q+1)Q} \sum\limits_{q =1}^{Q} \sum\limits_{a =0}^{q-1} f(\frac{a}{q})$$

This formula has the advantage of a clear sum indexation. But I am sure this option and other options exist in literature ? Any reference on the subject ?

With above formula it is intuitive (I think!) that for a bounded, continuous and differentiable function defined on $[0,1]$, we have:

$$ \int\limits_{\mathbb{Q}\cap [0,1]} f(x) dx = \int\limits_{0}^{1} f(x) dx$$

but it is not obvious to prove it. Any idea of a demonstration or reference with similar "sum - integral" equivalences is welcome.

(This post is linked to this post : On construction of a $\mathbb{Q}$ periodic function with Fourier series, in the way that I would like to use this integration for the type of functions defined in it.)

Answer 1

All you need is that $f$ is continuous on $[0,1]$.

Letting $$S(q) = \dfrac{1}{q} \sum_{a=0}^{q-1} f(a/q)$$ (which is a Riemann sum for $J = \int_0^1 f(x)\; dx$) and $$ R(Q) = \sum_{q=1}^Q \dfrac{2q}{Q(Q+1)} S(q)$$ you are defining $$ \int_{\mathbb Q \cap [0,1]} f(x)\; dx = \lim_{Q \to \infty} R(Q)$$

For any $\epsilon > 0$ there is $N$ such that $|S(q) - J| < \epsilon$ for $q \ge N$, and then for $Q > N$ $$ \eqalign{|R(Q) - J| &\le \sum_{q< N} \dfrac{2q}{Q(Q+1)} |S(q)-J| + \epsilon \sum_{q=N}^Q \dfrac{2q}{Q(Q+1)}\cr &\le \dfrac{N(N+1)}{Q(Q+1)}(\|f\|_\infty + |J|) + \epsilon } $$ which is less than $2\epsilon$ if $Q$ is sufficiently large. Thus indeed $\lim_{Q \to \infty} R(Q) = J$.