Definition of sigma algebra

  • #1

Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
1,761
123
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.
 

Related Threads for: Definition of sigma algebra

  • Last Post
Replies
1
Views
2K
Replies
3
Views
2K
Replies
5
Views
953
Replies
3
Views
4K
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
4
Views
926
  • Last Post
Replies
11
Views
668
  • Last Post
Replies
4
Views
1K
Top