Can the following construction be used to measure countable sets?

Solution 1:

I found a counter-example to the credibility of my measure that I should have found a long time ago. If we set $T_1=\left\{\left.\frac{m}{2n+1}\right|m,n \in \mathbb{Z}\right\}$ and $T_2=\left\{\left.\frac{v}{4r+2}\right|v,r\in\mathbb{Z} \right\}$, then according to my measure, $\mu(T_1)=\frac{2}{3}$ and $\mu(T_2)=1/3$.

However, if we change $T_1$ into $\left\{\left.\frac{2m}{4n+2}\right|m,n \in \mathbb{Z}\right\}$ we find that $T_1 \subset T_2$. Hence for my measure to have some credibility $\mu(T_1)<\mu(T_2)$. Morover $T_1$ had to be non-reduced to be comapared to $T_2$, which goes against my argument of simplification.

Hence, my measure cannot be used.

Hoefully someday there will be a measure that can be applied to any countable dense set.

What is the general equation for rotated ellipsoid?

$a+b=c \times d$ and $a\times b = c + d$

Summing $ \sum _{k=1}^{n} k\cos(k\theta) $ and $ \sum _{k=1}^{n} k\sin(k\theta) $

If the graph of a function $f: A \rightarrow \mathbb R$ is compact, is $f$ continuous where $A$ is a compact metric space?

Maximum area of triangle inside a convex polygon

What does actually probability mean?

Combinatorial sum identity for a choose function $\sum\limits_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$ [duplicate]

Are there any compact embedded 2-dimensional surfaces in $\mathbb R^3$ that are also flat?

Dense subset of two Banach spaces also dense in the intersection

Prove that $\int_{0}^{+\infty} u^{s-1} \cos (a u) \:e^{-b u}\:du=\frac{\Gamma(s)\cos\left(s\arctan \left(\frac{a}{b}\right)\right)}{(a^2+b^2)^{s/2}}$