Is the Definition of Sigma Algebra Limited to Countable Unions?

[AlonsoMcLaren]

1. Are uncountable unions of sigma algebras on a set X still a sigma algebra on X?

2. Are uncountable intersections of sigma algebras on a set X still a sigma algebra on X? (I think this statement is required to show the existence of sigma algebra generated by a set)

3. If 2 is true, can we use De Morgan's law to show than 1 is true?

4. If 1 is true, why do we require "countable unions" is the definition of sigma algebra?

 

[homeomorphic]

I think you mean uncountable union of sets, not uncountable union of sigma algebras because your number 4 there refers to the definition of sigma algebra which is concerned with countable unions of SETS, not countable unions of sigma algebras.

The reason why it's a countable union is because you want to be able to add it up. Countable means you can add up the measure of each set, It's not clear how to do that with an uncountable collection.