I don't know anything about the space of all distributions dual to smooth test functions, but do know a fair bit about computable measure theory (from a certain perspective).
First, you mention that you have a computable algorithm which generates a probability distribution. I believe you are saying that you have a computable algorithm from $[0,1]$ (or technically the space of infinite binary sequences) to some set $U$ where $U$ is the space of distributions of some type.
Say your map is $f$. How are you describing the element $f(x) \in U$? In computable analysis, there is a standard way to talk about these things. We can describe each element of $U$ with an infinite code (although each element has more than one code). Then $f$ works as follows: It reads the bits of $x$; from those bits, it starts to write out the code for the $f(x)$. The more bits of $x$ known, the more bits of the code for $f(x)$ known.
(Note, not every space has such a nice encoding. If the space isn't separable, there isn't a good way to describe each object while still preserving the important properties, namely the topology. Is say, in your example above, the space of distributions that are dual to smooth test functions, is it a separable space--maybe in a weak topology? Does the encoding you use for elements of $U$ generate the same topology?)
The important property of such a computable map is that it must be continuous (in the topology generated by the encoding, but these usually coincide with the topology of the space). Since $f$ is continuous, we know we can induce a Borel measure on $U$ as follows. If $S$ is an open set then $f^{-1}(S)$ is open and $\mu(f^{-1}(S))$ is known. Similarly, with any Borel sets, hence you have a Borel measure.
Borel measures are sufficient for most applications I can think of (you can integrate continuous functions and from them, define and integrate the L^p functions), but once again, I don't know anything about your applications.
Also, if the function $f$ doesn't always converge to a point in $U$, but only does so almost everywhere, the function $f$ is not continuous, but it is still fairly nice and I believe stuff can be said about the measure, although I need to think about it.
Update: If $f$ converges with probability one, then the set of input points that $f$ converges on is a measure one $G_{\delta}$ set, in particular it is Borel. The function remains continuous on that domain (in the restricted topology). Hence there is still an induced Borel measure on the target space. (Take a Borel set; map it back. It is Borel on the restricted domain, and hence Borel on [0,1]).
Update: Also, I am assuming that your algorithm directly computes the output from the input. I will give an example what I mean. Say one want to compute a real number. To compute it directly, I should be able to ask the algorithm to give me that number within $n$ decimal places with an error bound of $1/10^n$. An indirect algorithm works as follows: The computer just gives me a sequence of approximations that converge to the number. The computer may say $0,0,0,...$ so I think it converges to 0, but at some point it starts to change to $1,1,1,...$. I can never be sure if my approximation is close to the final answer. Even if your algorithm is of the indirect type, it doesn't matter for your applications. It will still generate a Borel map, albeit a more complex one than continuous, and hence it will generate a Borel measure on the target space. (The almost everywhere concerns are similar; they also go up in complexity, but are still Borel.) Without knowing more about your application it is difficult for me to say much specific to your case.
Am I correct in my understanding of your construction, especially the computable side of it? For example, is this the way you describe the computable map from $[0,1]$ to $U$?
On a more general note, much of measure theory has been developed in a set theoretic framework. This isn't very helpful with computable concerns. But using various other definitions of measures, one is able to once again talk about measure theory with an eye to what can and cannot be computed.
I hope this helps, and that I didn't just trivialize your question.
Best Answer
I found Bruce Bartlett's MSc dissertation Categorical Aspects of Topological Quantum Field Theories a very clear and well-written introduction to TQFTs and related matters.