Conditioning Probability on a Language With Measure 0
I don't know of a general approach to handle this, but in the case of $\omega$-regular languages, this has been done.
One approach, which I think is first introduced in the paper Computing Conditional Probabilities
in Markovian Models Efficiently is the following:
Given the language $L$, let $D$ be a DPW for it. Now, start by constructing a Markov chain $M$ that captures your distribution on $\Sigma^\omega$ (uniform, in your case).
Then take the product of $M$ and $D$.
Now, in the product MC, you have the property that the probability that the run is accepting is exactly the probability of $L$ (0, in your case, but it works for any $L$).
We now modify the product by &quot;redistributing&quot; the probability: whenever the MC reaches an end-component (ergodic component) in which the probability of acceptance is 0, the run restarts in the initial state. This ensures that all $0$-probability-of-acceptance runs get &quot;another chance&quot;.
You can use this to essentially restrict yourself only to words in $L$, and now you can ask what the probability of another language is, in this MC, and this provides a reasonable definition of conditional probability.
I'm slightly sketchy on the details, but I think this should work.
I've used this technique in a paper here: https://arxiv.org/abs/1608.06567.