Mathematicians like to ask covering various sets with open intervals and the answers to these riddles have strange tendencies to become strange lemma's or theorems Heine-Borel Theorem, lebesgue's Number Lemma, Vitali Covering Lemma, Besicovitch Covering Theorem, (etc)
By open interval he means the stretch of real numbers: $a<x<b$.
My challenge is to strictly weighted-ly cover all the rational numbers between $A$ and $B$ with open intervals. By weighted-ly cover means each paticular rational number lies in a particular interval. However, each interval has a weight $a_i$. The challenge here is that weighted sum must be less than $k'$.
Before we proceed let's return to normal measure theory where $a_i=1$ for all weights. Then we have $k' = \int_{A}^B dx= B-A$.
Now, let us look at limit of a sum as Riemann integral:
$$ {\displaystyle \Delta x={\frac {B-A}{n}}}$$
$$ A + r \Delta x \to x$$
OR
$$ r \to \frac{x -A}{\Delta x}$$
Note: $r$ is a dummy variable.
Now, in the limit $\Delta x \to 0$
$$ \int_A^B f(x) dx = \lim_{\Delta x \to 0}\Delta x\left[f(a+\Delta x)+f(a+2\,\Delta x)+\cdots +f(b-\Delta x)\right]= \lim_{\Delta x \to 0}\sum_{r=1}^n f(a+ r \Delta x) \Delta x$$
So what will $k$ be in the case where all $a_i$ aren't the same? Let's put this question in light of the answer:
Claim: If $\lim_{n \to \infty} \frac{\log^2(n)}{n}\sum_{r=1}^n |b_r| = 0$ and $f$ is smooth, then $$\lim_{k \to \infty} \lim_{n \to \infty} \sum_{r=1}^n a_rf\left(\frac{kr}{n}\right)\frac{k}{n} = \left(\lim_{s \to 1} \frac{1}{\zeta(s)}\sum_{r=1}^\infty \frac{a_s}{r^s}\right)\int_0^\infty f(x)dx.$$
where $a_r = \sum_{e|r} b_e$
All we now do is add the notation:
$$ \int_{A}^B a_{\frac{x-A}{dx} } f(x) dx= \lim_{k \to B-A} \lim_{n \to \infty} \sum_{r=1}^n a_rf\left(A+\frac{kr}{n}\right)\frac{k}{n}$$
In fact using the co-ordinate transformation $dy = f(x) dx$ with a function. We also define a coordinate transformation or mapping $ g(x) = y$. Hence,
$$ r \to \frac{x-A}{dy} \frac{dy}{dx}= f(x) \frac{x-A}{dy} = f(g^{-1} (y)) \Big( \frac{g^{-1} (y)- A}{dy} \Big) $$
This enables us to talk about coordinate transformations.
Question
What is the measure of $\int_{A}^B a_{\frac{x-A}{dx} } f(x) dx$ ?
If $$\lim_{n \to \infty} \frac{\log^2(n)}{n}\sum_{r=1}^n |b_r| = 0$$ and $$a_r = \sum_{e|r} b_e$$
