In the definition of a measure space (or a probability space), the measurable sets are required to form a $\sigma$-algebra. That is, they must be closed under complements and countable unions (and thus countable intersections). Does anyone know why these closure properties were chosen?
I see a good argument for closure under complements: if the entire space has measure $r\in\mathbb{R}_{\geq 0}\cup\{\infty\}$, we would like $\mu(A^C)=r-\mu(A)$. I also see a good argument for closure under disjoint unions (and perhaps countable disjoint unions), since we would like $\mu(A\cup B)=\mu(A)+\mu(B)$, when $A\cap B=\emptyset$.
However, I'm not clear why we assume closure under countable unions or intersections (or even finite ones) . If we think of non-measurable sets as pathological, is there a reason to expect that the union and intersection of two non-pathological sets will be non-pathological?
EDIT: The first answer points out that a family of sets is a $\sigma$-algebra if and only if it is a $\lambda$-system and it is (or is generated by) a $\pi$-system (by Dynkin's theorem). Since we already have arguments for why measurable sets should form a $\lambda$-system, the original question can be considered equivalent to: "why do we assume measurable sets form a $\pi$-system"?
You want these intuitive properties for a probability:
The probability of a countable union of disjoints events is the sum of it probabilities.
The probability of the complementary event is the "complementary probability".
These two properties essentially defines a probability function over a collection of sets. A collection of sets that is closed under complementation and countable union of disjoint sets is called a $ \lambda $-system (it is also called a Dynkin system).
But it can be shown that the induced $\lambda $-system over a collection of sets that is already closed by finite intersections defines a $\sigma$-algebra. This theorem is called sometimes as the $ \lambda -\pi$ theorem (because a system closed by finite intersections is called a $\pi$-system).
Because its natural to ask by the probability of the intersection of events (that is, the event that represent that two events happens "at the same time"), or it union (that is: if at least one of them happen) then its natural to define a $ \sigma $-algebra and a probability function such that the two previous properties at the top of this answer holds. Then everything follows naturally from there.