Can you have perfect secrecy with countable message/key spaces by dropping countable additivity?
Let the key, message, and ciphertext spaces all be $\mathbb{Z}$, which is in bijection with $\mathbb N$. I'm going to pick a specific bijection by making the non-negative integers map to evens and the negative integers map to odds, and let $\mu$ be the asymptotic density measure applied to $\mathbb{Z}$ using this bijection. We can then construct a one-time pad by setting $c = m + k$. For any (possibly finitely-additive) probability measure $\mu_M$ chosen by the adversary on $m$, and any set $Y \in F$, $(\mu_M \times \mu)(\{(m, k) \in \mathbb Z^2 \mid m + k \in Y\}) = \mu(Y)$. That is, the resulting measure on the ciphertext is uniform, no matter what measure $m$ comes from. I am stating it this way so as to avoid the division inherent in the $P(m \in X | c \in Y)$ expression in your question.
Details: First, we need to show that $\mu$ is translation invariant on $\mathbb Z$. $\mu(A)$ could equivalently be defined as $\lim_{n \to \infty} \frac{A \cap [-n, n]}{2n + 1}$. To see that translation invariance holds, notice that $|[-n, n] \cap A|$ and $|[-n, n] \cap (A - x)|$ only differ by at most $2 x$, since $[-n, n]$ and $[-n + x, n + x]$ overlap everywhere except for $x$ points at either end. This becomes irrelevant in the limit as $n \to \infty$.
Next, we need to evaluate the product measure. Unfortunately product measures do not seem to be very well defined for finitely additive measures. I will pick one product measure, but in general it is not unique.
\begin{align*}
(\mu_M \times \mu)(\{(m, k) \in \mathbb Z^2 \mid m + k \in Y\})
&amp;= \int_{\mathbb Z} \mu(\{k \in \mathbb Z \mid m + k \in Y\}) d\mu_M(m) \\
&amp;= \int_{\mathbb Z} \mu(Y - m) d\mu_M(m) \\
&amp;= \int_{\mathbb Z} \mu(Y) d\mu_M(m) \\
&amp;= \mu(Y) \\
\end{align*}
Here we have used that $\mu$ is translation invariant on $\mathbb Z$ and that $\mu_M$ is a probability measure, so the integral (i.e., expected value) of a constant is that constant.
However, none of this is physically realizable, at least with a digital computer. Any way of generating a random key is going to give a distribution that has countable additivity, since it can always be broken down into discrete choices, with each outcome having paths leading to it with only have a finite number of choices. I don't think it is physically possible in general, but I have less justification for that.