About halfway through "A frequentist understanding of sets of measures" by Fierens, Rêgo, and Fine (pdf available here) I encountered the claim that "there is a recursive probability measure such that ..." (p. 1182). "Recursive measure" is used in two other places in the same paper. I know what a measure is, I know the general concept of recursion, and I kind of know what a recursive function is in the computability sense. I also kind of know enough topology to understand the rest of the sentence in which this phrase occurs (see below). But what is a recursive measure? I can't figure out what this might mean.
In case it's useful, here's the context: This is a mathematical paper by engineers discussing something like a discrete time random process, where at each time $t$ a selection is made from a set $\cal M$ of probability measures, and the chosen measure is used to generate the next outcome in the process. The measure is selected by a function of past outcomes. There is a metric on $\cal M$. The full sentence in which "recursive measure" first appears reads:
Assume that there is an $\epsilon$-cover of $\cal M$ by $N_e$ open balls with centers in the set $M_e = \{\mu_1, \mu_2, \ldots, \mu_{N_\epsilon}\}$ such that, for each $\mu_i$, there is a recursive probability measure $\nu \in B(\epsilon, \mu_i) \cap \cal M$.
$B(\epsilon, \mu_i)$ is an open ball with radius $\epsilon$ and center $\mu_i$, and $\epsilon$ is a number greater than 1 over a large integer.
I just give two examples, there may be better explanations...
An example of a recursive measure $\mu$ on $\mathbb R$ : Consider a recursive (easy) bijection $\alpha_i$ from $\mathbb N$ to $\mathbb Q$ and define $$\mu(\{\alpha_i\})=2^{-(i+1)}$$
Hence $\mu(\mathbb R)=1$ and to compute a good approximation of $\mu(S)$ for any $S\subset\mathbb R$, just test if $\alpha_i\in S$ for $0\le i\le n$ to obtain an approximation of $\mu(S)$ with precision $2^{-n+1}$.
If you have a good description of $S$ it should not be hard to compute if $\alpha_i\in S$ or not.
Now consider another example of measure $\nu$ such that
$$\nu(\{\alpha_i+\Omega_F\})=2^{-(i+1)}$$
where $\Omega_F$ is the Chaitin's constant (it could be any non recursive real). It's just a shifted version of $\mu$.
Then there is no algorithm that can compute good approximation of $\nu$ on sets (because it would imply that you can compute good approximations of $\Omega_F$, and you can't).