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In probability, events are considered to be closed under countable union and complement, so mathematically they are modeled by $\sigma$-algebra. I was wondering why events are considered to be closed under countably union and complement?

In Nate Eldredge's post, he has done an excellent job on explaining this, by using whether questions are answered or not as an analogy to whether events occur or not, if I understand his post correctly. However, if someone could explain plainly without analogy, it could be clearer to me.

I was particularly curious why events are not considered to be closed under infinite (possibly uncountably) union, but instead just under countably union? So possibly to model events using the power set? I think this is not addressed in Nate Eldredge's post.

My guess would be that the reason might be related to the requirement on the likelihood of any event to occur to be "measurable" in some sense. But how exactly to understand this requirement is unclear to me.

PS: This post is related to my previous one Interpretation of sigma algebra, but the questions asked in these two are not the same.

Thanks and regards!

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up vote 4 down vote accepted

As Jonas mentioned, allowing arbitrary unions is not "consistent", in the sense that there is no proper definition of probability. This is also related to the fact that infinite sums make much more sense when countable, since it's not clear how to attach a finite number to an uncountable sum of positive reals.

On the other hand, many desirable events are describable using countable unions and intersections. For example, events like "the random walk returns to the origin" is a union of countably many events "the random walk returns to the origin at time $t$", and any one of those is a finite union of "basic" events.

In general, first order properties always correspond to taking countable unions and intersections; this means that if you have a statement of the form "$\forall x \exists y \cdots P(x,y,\ldots)$", where $x,y,\ldots$ are integers, and the $P$s are basic events (e.g. for a random walk, depend on finitely many times), then the corresponding event is guaranteed to be in the $\sigma$-algebra, i.e. is guaranteed to have assigned to it a "probability".

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Spot on. Re uncountable sums of positive reals, it is clear that there is no way to attach anything else than $+\infty$ to any of these. Namely: let $(x_t)_{t\in T}$ denote any family of positive real numbers $x_t$ indexed by an uncountable set $T$. Then, for every $M$, there exists a finite subset $S$ of $T$ such that the sum of $x_t$ over $t\in S$ is greater than $M$. – Did Feb 26 '11 at 20:36

If events were closed under arbitrary unions, and if singletons are events, then everything would be an event, because every set is the union of the singletons it contains.

Trying to define probabilities or measures on all sets can lead to problems.

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