How to show something is a sigma-algebra

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In summary, the conversation is about a specific example of sigma-algebra, which is a collection of subsets of a set X that follows certain properties. The example given is the sigma-algebra generated by singletons, which means that all the subsets are either countable or their complements are countable. The conversation also discusses the properties of sigma-algebra, which include being non-empty, closed under complements, and closed under countable unions. The person initially had trouble understanding the example, but eventually figured it out.
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Homework Statement


I was reading this Wiki article: http://en.wikipedia.org/wiki/Sigma-algebra and don't quite understand one of the examples.

"The collection of subsets of X which are countable or whose complements are countable (which is distinct from the power set of X if and only if X is uncountable.). This is the σ-algebra generated by the singletons of X."


Homework Equations


1. Σ is not empty,
2. Σ is closed under complements: If E is in Σ then so is the complement (X \ E) of E,
3. Σ is closed under countable unions: The union of countably many sets in Σ is also in Σ.

The Attempt at a Solution


I kind of understand sigma-algebra, but I really don't get this example... If it's the sigma-algebra generated by singletons, then how can the first property be satisfied?
 
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Nevermind, I figured it out. :)
 

1. What is a sigma-algebra?

A sigma-algebra is a collection of sets that satisfies certain properties, namely closure under countable unions and complements. It is used in measure theory to define measurable sets and measure functions.

2. How do you show that a set of subsets is a sigma-algebra?

To show that a set of subsets is a sigma-algebra, you need to prove that it satisfies the two properties of closure under countable unions and complements. This can be done by showing that the set is closed under countable unions and that every set in the collection has a complement in the collection.

3. What is the importance of sigma-algebras in mathematics?

Sigma-algebras are essential in measure theory as they allow us to define measurable sets and measure functions. They are also used in probability theory to define probability spaces and random variables.

4. Can you give an example of a sigma-algebra?

One example of a sigma-algebra is the Borel sigma-algebra on the real numbers, which is the smallest sigma-algebra that contains all open intervals on the real line. This sigma-algebra is used in probability and statistics to define measurable events and random variables.

5. How is a sigma-algebra different from a sigma-field?

A sigma-algebra and a sigma-field are essentially the same thing, with the only difference being that a sigma-field is defined on a non-empty set, while a sigma-algebra is defined on any set. In practice, the terms are often used interchangeably.

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