This is a bit of a soft question because I'm not very strong in computability and recursion theory, but here goes.
In measure theory, one starts with a measurable space $(\Omega, \mathcal F)$, which is just a set $\Omega$ and a sigma-field $\mathcal F$ of subsets of $\Omega$. It is often convenient to assume that $\mathcal F$ is countably generated, which means that there is some countable subset $\mathcal C$ of $\mathcal F$ such that $\mathcal F$ is the smallest sigma-field containing $\mathcal C$.
My (again, soft) question is whether countable generation can be motivated by or viewed as a kind of computability condition.
My (very rough) idea is something like the following: the members of $\mathcal F$ should be "computable" (I don't have a precise definition in mind here, just a rough idea) and computability (whatever it is) requires countable operations, so $\mathcal F$ should be countably generated.
Perhaps an analogy will help. A real number is computable if it can be computably approximated by the members in a countable set (the rationals). That sentence can be made perfectly precise using computability and recursion theory. I want to know if the sentence "A member of $\mathcal F$ is computable if it can be computably approximated by the members of a countable set ($\mathcal C$)" can be made perfectly precise in way that entails that $\mathcal C$ generates $\mathcal F$.
Thanks for your patience with this very open-ended question!
Countable generation of the sigma algebra $\mathcal{F}$ implies that for any $\sigma$-finite measure $\mu$ on $\mathcal{F}$ and $p \in [1, \infty)$, $L^p(\Omega, \mathcal{F}, \mu)$ is separable, i.e. has a countable dense set.
Another related result, which is actually part of the proof of the previous result is that if $\mathcal{A}$ is an algebra that generates the $\sigma$-algebra $\mathcal{F}$ and $\mu$ is a finite measure on $(\Omega, \mathcal{F})$, then the closure of $\mathcal{A}$ in $L^1(\Omega, \mathcal{F}, \mu)$ contains $\mathcal{F}$, that is, for any $A \in \mathcal{F}$ and any $\varepsilon > 0$, there exists $B \in \mathcal{A}$ such that $\|1_{A} - 1_{B}\|_{L^1} = \mu(A \Delta B) < \varepsilon$. This result is an immediate consequence of the $\pi$-$\lambda$ theorem since $\mathcal{D} = \{A \in \mathcal{F} : A \in \overline{\mathcal{A}}\}$ is a $\lambda$-system.
Both of these results are very useful in probability theory.