Constructing a Measure on the Rational Numbers I was wondering if there is any known examples of measures on the set of rational numbers besides Lebesgue measure.  In particular, an example of a probability measure on $\mathbb{Q}$ would be nice to see.  
Here is a somewhat naive attempt:  Let $X = \mathbb{Q}, \Sigma = 2^X $, the sigma algebra which is the set of all subsets of $\mathbb{Q}$.  Now for $E \subset \Sigma$ define $m(E) = \lim_{n \to \infty} \frac{|E \cap \{ q_1, q_2, ..., q_n \}|}{n}$ where $(q_j)_{j=1}^\infty$ is an enumeration of the rationals and $|A|$ denotes the cardinality of a finite set $A$.  Is this a probability measure?
 A: A simple example of such a measure is the following, but this is a boring example.
Let $\{ q_n \}_{n=1}^ \infty$ be an enumeration of the rationals. 
Define
$$m=\sum_{k=1}^\infty \frac{1}{2^k} \delta_{q_k} \,.$$
This measure can be defined alternatelly
$$m(E) = \sum_{ q_n \in E} \frac{1}{2^n}$$
A: No, your suggestion is not a probability measure as it is not additive over countable unions of disjoint sets.  
If $t$ is a natural number then $m(\{q_t\})= \lim_{n \to \infty} \frac1n=0$ and so $\sum_t m(\{q_t\})=0$. But $m( \mathbb{Q})= \lim_{n \to \infty}\frac{n}{n}=1$.
You need a discrete measure. 
For example $\mathbb{Q} \cap (0,1)$ you could use $m \left( \left\{ \frac{a}{b} \right\}\right) =\frac{\zeta(k)}{\zeta(k-1) - \zeta(k) } \left(\frac{1}{b}\right)^k$ for some $k\gt2$ and $a$ and $b$ coprime; this can be extended to all rationals. 
A: Note that $m({q}) = 0$ for every $q \in \mathbb{Q}$, whereas $m(\mathbb{Q}) = 1$, so $m$ cannot be $\sigma$-additive. This argument shows that there are no probability measures on a denumerable set that assign the same measure to every singleton.
Moreover, the limit in your definition does not always exist.
There are, however, interesting examples of finitely additive measures (or probability charges). The example you mention if in fact to being a probability charge, but one must be careful enough to consider the cases where the limit does not exist see this article.
