Formally assigning probabilities to sample spaces that change over time As mentioned in the title, how are probabilities assigned to changing sample spaces?
To make my question more clear, take the classic Polya's Urn Scheme, where we have an urn containing $b$ black balls and $r$ red balls. After each draw of a ball from the urn, the ball is replaced with $c$ more balls of the same color back into the urn. Regarding the first draw, I'm clear..since each ball is equally likely to be picked, a uniform probability space $(\Omega, \mathcal{F}, \mathbb{P})$ does the job once $\Omega$ is taken to be the set of all balls in the urn.
After the first draw, or any draw for that matter, the set of balls in the urn is no longer the same as $\Omega$. But any solution to this problem that I've seen does not mention this..and the answer is computed directly. Of course, its intuitively clear what's going on..but is there a more formal way of depicting this? Do we move to a new sample space $(\Omega', \mathcal{F'}, \mathbb{P}')$ after each draw?
What is the general way of working with sample spaces that change over time / after the occurrence of some events?
 A: What I would do is the following:
Fix a probability space $(\Omega,\mathcal{F},P)$ that captures all the states of the world along the urn process. On that space define two series of random variables $B_n$ and $R_n$ to model the number of black and red balls after $n$ steps in the process (e.g. $B_0=b$ $P$-a.s.).
In this setting one would be able to deduce recursive relations for $B_n$ and $R_n$ without changing the underlying probability space. Instead one gets two series of new sample spaces induced from the random variables. Assuming e.g. all of the $B_n$ are real valued, one would now sample in $(\mathbb{R},\operatorname{Borel}(\mathbb{R}),P^{B_n})$. And these spaces might change in $n$.
Each $\omega\in\Omega$ defines a random series of balls drawn from the urn. The specific definition of $\Omega$ and $P$ do not matter, as long as the induced measures in the sample spaces are as expected (e.g. $P^{B_n}$ could be some discrete distribution conditionally depending on $B_{n-1}$ and $R_{n-1}$).
Edit:
To address the question more directly:
With the above approach we do not have to change anything about $\Omega$ in the process. Instead, we define random variables on $\Omega$ and have their induced sample spaces change with $n$ (though it’s really only the induced measure, e.g. $P^{B_n}$, that changes in $n$).
A: I think the typical way to do this would be to set $\Omega := \{B,R \}^{\mathbb{N}}$, i.e. $\Omega$ is the set of all sequences of black and red ball draws.  Then $\mathcal F_n$ would be the filtration generated by the first $n$ draws, and $\displaystyle \mathcal F = \mathcal F_\infty = \bigvee_{n=1}^{\infty} F_n$.
Defining the probability measure $\mathbb{P}$ is slightly trickier, and relies on the Kolmogorov extension theorem.  Essentially we define $\mathbb{P}_n$ on $(\Omega, \mathcal F_n)$ to be the correct marginal distribution for the first $n$ tosses, so if $\omega = \omega_1\omega_2 \cdots \omega_n \cdots \in \Omega$ then $\mathbb{P}_n$ is the probability of getting the sequence $\omega_1\omega_2\cdots \omega_n$ in the first $n$ tosses.  That probability is probably slightly difficult to compute, but can be done recursively.  Then the Kolmogorov extension theorem guarantees the existence of a probability measure $\mathbb P$ on $(\Omega, \mathcal F)$ that agrees with the $\mathbb{P}_n$s on each $\mathcal F_n$.  Specifically, if $A \in \mathcal F_n$ then $\mathbb P(A) = \mathbb P_n(A)$.
So our probability space is $(\Omega, \mathcal F, \mathbb{P})$, and we keep the same $\Omega$ throughout.  This way we don't really have the probability space changing, because we think of the probability space as the entire sequence of draws.  The way we keep track of what we intuitively might think of as the "probability space changing" is with the filtrations $\mathcal F_n$ and the finite marginal probabilities $\mathbb{P}_n$.  The $\mathcal F_n$ represent the information we have after seeing $n$ draws, and the $\mathbb{P}_n$ assign probabilities to events that are observable within the first $n$ draws.  I think part of the reason this is confusing is because in order to actually do a Polya's Urn Scheme in the real world you would probably have to be changing what you're drawing from each time, but that's different from the way it is modeled.  That's because it would be quite complicated to model a randomly changing probability space, and we would have to do a lot more work to use theorems like Fatou's lemma or the dominated convergence theorem.
EDIT: For a simple example, suppose we only draw from the urn twice and we start with $1$ black ball and $2$ red balls and add $1$ ball at each step.  Then $\Omega = \{BB,BR,RB,RR\}$ is the set of possible sequences and serves as our sample space, our $\sigma$-algebras $\mathcal F_n$ (basically the elements we can tell apart after observing the first $n$ draws) are
\begin{align*}
\mathcal F_0 &= \{\emptyset, \Omega\} \\
\mathcal F_1 &= \{\emptyset, \Omega, \{BR,BB\}, \{RB,RR\} \} \\
\mathcal F_2 &= 2^\Omega.
\end{align*}
Our probability measure $\mathbb P$ is found by finding the probability of each sequence, so for example $\mathbb P(\{BB\}) = \frac 13 \cdot \frac 12 = \frac 16$.  If you wanted to, you could also define a probability measure $\mathbb{P}_1$ on $\mathcal F_1$ corresponding to the probability of the first draw, so
\begin{align*}
\mathbb{P}_1(\{BR,BB\}) &= \frac 13 \\
\mathbb{P}_1(\{RB,RR\}) &= \frac 23.
\end{align*}
