Markov chains and conditioning on impossible events Consider a Markov chain $(X_0,X_1,\ldots)$ with a state space $S\equiv\{s_1,s_2\}$ and the following matrix of “transition probabilities” (I will explain the use of quotation marks below):
\begin{align*}
\begin{array}{c|cc}
&s_1&s_2\\
\hline
s_1&1&0\\
s_2&1&0
\end{array}
\end{align*}
That is, no matter what initial state the system starts in, it will always end up in state $s_1$ in one period and stay there forever.

Rigorously speaking, these “transition probabilities” are to be interpreted as follows:
\begin{align*}
\mathbb P\,(X_{n}=s_1\,|\,X_{n-1}=s_1)=&\,1,\\
\mathbb P\,(X_{n}=s_2\,|\,X_{n-1}=s_1)=&\,0,\\
\mathbb P\,(X_{n}=s_1\,|\,X_{n-1}=s_2)=&\,1,\\
\mathbb P\,(X_{n}=s_2\,|\,X_{n-1}=s_2)=&\,0
\end{align*}
for each $n\in\mathbb N$.

My concern is that the last two probabilities are ill-defined (except possibly for $n=1$), because for any given initial probabilities, the condition events $\{X_{n-1}=s_2\}_{n=2}^{\infty}$ have zero probability! Strictly speaking, therefore, the above matrix cannot be interpreted as conditional probabilities because of the problem of conditioning on events that never occur.

What is the standard resolution of this technical problem? Does one make the hand-waving assumption of defining conditional probabilities that depend on impossible events anyway, or is there a more sophisticated and rigorous way around this issue?

Any input is appreciated.
 A: In fact one  defines the transition kernels:
$$K(\omega, j) = \Bbb{E}(1_{X_n = j} \mid \mathcal{F}_{n-1})(\omega) \quad \Bbb{P } \,a.s. $$ 
This means that we have a regular conditional probability that allows us to talk about the jumps of our process.
The markov property consists in saying that $K(\cdot, j)$ is $\sigma(X_{n-1})$ measurable
that is 
$$K(\omega, j) = \phi_j(X_{n-1})(\omega) \quad \Bbb{P}\, a.s.  $$
Hence there is no hand waving.
A: As you point out, there is no guarantee that each of the states
in a Markov Chain can be visited at any particular step $n$.
Another good example with lots of impossible visits is a periodic
chain where any one state can only be visited on every $d$th step,
where $d$ is the period.
More subtle are examples on countably infinite state spaces.
For example, consider a random walk on the non-negative integers
with a 'drift' towards 0 and a reflecting barrier at 0. The
higher-numbered states may be visited extremely rarely.
Some texts do not address the interpretation of conditional
probability for conditions with probability 0 at all. Others
agree to ignore the conditional probability in cases where
it is moot. Yet others, instead of your "for all $n$" seem
to say "for all applicable $n$." In any case, this issue is not
a difficulty in applications.
