Proving Infinite Sigma-Algebra's Countable Disjoint Subsets

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In summary, the conversation discusses the possibility of a sigma-algebra being both infinite and countable. The speaker suggests that if a sigma-algebra has a countable number of disjoint subsets, then it cannot be countable. They also consider the idea of a sigma-algebra consisting of an infinite number of subsets and the need to show that it has a countably infinite number of disjoint subsets. The conversation ends with a suggestion of using proof by contradiction to show this.
  • #1
Zaare
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I'm supposed to answer the question "Can a sigma-algebra be infinite and countable?"
I think I can show that if it has a countable number of disjoint subsets, then it can't be countable considering the possible combinations of the subsets.
Now I need to show that if a sigma-algebra consists of an infinite number of subsets, then it has a countable number of disjoint subsets.
Any ideas on how I can do this?
 
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  • #3
I think I can show that if it has a countable number of disjoint subsets, then it can't be countable considering the possible combinations of the subsets.

No you can't.

Now, if you instead said countably infinite... :smile:


Now I need to show that if a sigma-algebra consists of an infinite number of subsets, then it has a [countably infinite] number of disjoint subsets.

(I edited it)

Proof by contradiction, maybe?
 
  • #4
Hurkyl said:
No you can't.

Now, if you instead said countably infinite... :smile:

That's what I meant. I was sloppy. :redface:
Thank you both for the help.
 

What is an infinite sigma-algebra?

An infinite sigma-algebra is a collection of sets that satisfies certain properties, such as closure under countable unions and complements. It is used in measure theory to define the concept of a measurable set.

How is an infinite sigma-algebra different from a finite sigma-algebra?

An infinite sigma-algebra is a collection of sets that contains an infinite number of elements, while a finite sigma-algebra contains a finite number of elements. In other words, the number of sets in an infinite sigma-algebra is uncountable, while the number of sets in a finite sigma-algebra is countable.

What are the applications of infinite sigma-algebras?

Infinite sigma-algebras are used in measure theory to define the concept of a measurable set, which is essential in probability theory and statistics. They are also used in mathematical analysis to study the convergence of sequences and series.

Can an infinite sigma-algebra contain both countable and uncountable sets?

Yes, an infinite sigma-algebra can contain both countable and uncountable sets. It is not limited to a specific type or size of sets, as long as it satisfies the properties of a sigma-algebra.

What are some examples of infinite sigma-algebras?

One example of an infinite sigma-algebra is the Borel sigma-algebra, which is the smallest sigma-algebra that contains all open sets in a given topological space. Another example is the Lebesgue sigma-algebra, which is used to define Lebesgue measure on the real numbers.

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