I've got 2 questions regarding probability spaces/events/sigma algebras etc.

1. If you got the event {1,2,3}, you could see this is as 1 or 2 or 3, right ? Is this the standard notation for events ?

How would you notate the event 1 and 2 and 3 ?

2. Why would you take the set of events to be smaller than all the subsets of the sample space?In my book, they are talking about the properties of a sigma algebra field.

And if the set of events are all the subsets of the sample space this makes sense to me.

But I'm like, why would bother about smaller sets.

For example:

If you throw a dice, and you got the sample space {1,2,3,4,5,6}.

I could just only put the zero set and the sample space in my set of events...

But I'm like, why would you only look at those events ?

And why can I look at some smaller sets, like the trivial one, but not at the set of the events {2,4,6} and {1,2,3,4,5,6}?

I mean, I can see that this is not a sigma field. But to me the trivial sigma field is as useless as set with the events {2,4,6} and {1,2,3,4,5,6}.

Anyone here who can help me out a little bit ?