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If I wanted to compare the densities of sets of numbers over the positive integers, what measures are available to me? I have some particular properties in mind.

Let $X_n=\{(2n-1)2^m:m\in\mathbb{N_{>0}}\}$

Let $X=\{X_n:n\in\mathbb{N_{>0}}\}$

So e.g. $X_3=\{3,6,12,24,48,...\}$

Clearly $$\bigcup_{X_n\in\ X}X_n=\mathbb{N_{>0}}$$

I want a measure $M_n$ of the density of $X_n$ in $\mathbb{N_{>0}}$ satisfying various properties:

It is always greater if another $X_n$ is added to the union.

Every $X_n$ has a measure

The measure of $X_n$ is smaller, for bigger $n$.

The measure of any subset of $\mathbb{N_{>0}}$ is finite

Just using some basic school maths with no prior experience of measures of density, it would appear to me that this is appropriate:

$$M_n=\frac{1}{2^n}$$

And for some collection $X_N\subset X$:

$$M_N=\sum_{X_n\in X_N}M_n$$

Is there anything I am doing badly wrong here, any shortcomings, or any clear improvement I could make?

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    The elementary notion of *asymptotic density* allocates to every subset $A$ such that the limit exists, the measure $$d(A)=\lim_{n\to\infty}\frac{\# (A\cap[1,n])}n$$ Then, for every $A$ such that $A$ exists, $0\leqslant d(A)\leqslant1$, for every $A\subseteq B$ such that $d(A)$ and $d(B)$ both exist, $d(A)\leqslant d(B)$, for every finite $A$, $d(A)=0$, and, for every $n$, $d(X_n)=0$. Two major defects of $d$, which makes that people are using other measures of subsets of $\mathbb N$, are that $d$ is not sigma-additive, naturally, and, perhaps less obviously, ... – 2017-03-25
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    ... that $d$ is not behaving well with respect to finite intersections in the sense that, even if $d(A)$ and $d(B)$ both exist, it may happen that $d(A\cap B)$ does not. This example is presented in quite a few first courses in probability theory. – 2017-03-25
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    Four days later... quite impressive (absence of) reaction from the OP. Too much mathematics in there, perhaps? – 2017-03-29

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