Rational probabilities A probability space is generally defined to be a sample space $\Omega$ with a sigma algebra $F\subseteq 2^{\Omega}$ and a probability function $P:F\to \mathbb{R}$.
Is there anything inconsistent about taking the probability function to be $P:F\to\mathbb{Q}$ instead of $P:F\to \mathbb{R}$? That is, does it make sense to think of probabilities as rationals rather than reals?
Generalizing slightly, for what spaces $X$ can you consider a function $P:F\to X$ and have the resulting system be something that we would recognize as a probability space?
 A: Expanding on the answer of Tom Ellis, one can show that such a probability can have only finitely many values.
To see this, suppose P takes infinitely many values and take a set $A\in F$ such that $0<P(A)<1$. It has to be the case that $P$ takes on infinitely many values on $\{B\in F:B\subseteq A\}$ or on $\{B\in F:B\subseteq A^C\}$. In the first case let $M_1=A$ and otherwise let $M_1=A^C$. Now pick $A'\subseteq M_1$ with $0<P(A')<P(M_1)$ and continue this way to get a decreasing sequence of measurable sets $M_1\supset M_2\supset\ldots$ with $P(M_1)>p(M_2)>\ldots$. For all $n$, let $F_n=M_n\backslash M_{n+1}$. Then $(F_n)$ is a seuence of disjoint measurable sets with positive probability. You can find a subsequence $(F_{n_m})$ such that $P(F_{n_{m+1}})<P(F_{n_m})/2$ for all $m$. Now for every set of natural numbers $I\subseteq\mathbb{N}$, $\sum_{m\in I} P(F_{n_m})$ is a different real number. Now there are only countably many rational numbers but uncountably many subsets of $\mathbb{N}$, so $P$ cannot have only rational values.
This answer makes of course ample use of countable additivity.
A: A measure ("probability function") has to satisfy various properties, among them countable additivity.  In particular this implies that if $A_n$ is an increasing sequence of elements of $F$ then $P(A_n) \nearrow  P(\cup A_n)$.
This condition means that there are very few interesting measures that take values only in $\mathbb Q$.  I imagine that the only ones that exist are those on a $\sigma$-algebra that is in some sense "discrete".
