Is the Definition of Sigma Algebra Limited to Countable Unions?

In summary, the conversation discusses the properties of sigma algebras on a set X, specifically whether uncountable unions and intersections of sigma algebras on X are still sigma algebras. It is suggested that the existence of a sigma algebra generated by a set may require the statement to be true. The use of De Morgan's law is also considered to prove the truth of 1. The group concludes that the definition of a sigma algebra involves countable unions instead of uncountable ones due to the difficulty of adding up the measure of an uncountable collection.
  • #1
AlonsoMcLaren
90
2
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?
 
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  • #2
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.
 

What is a sigma algebra?

A sigma algebra, also known as a sigma-field, is a collection of subsets of a given set that satisfies certain properties. It is often used in the context of measure theory to define a measure on a set.

What are the properties of a sigma algebra?

A sigma algebra must contain the empty set and the entire set, and it must be closed under countable unions and complements. It is also closed under countable intersections if it is a finite or countably infinite sigma algebra.

How is a sigma algebra different from a standard algebra?

A sigma algebra is a more general concept than a standard algebra. While a standard algebra is only closed under finite unions and intersections, a sigma algebra is closed under countable unions and intersections. In other words, a sigma algebra has more flexibility in terms of the number of sets in an operation.

How is a sigma algebra used in measure theory?

In measure theory, a sigma algebra is used to define a measure on a set. By defining the subsets of a given set that are "measurable" with respect to the measure, a sigma algebra allows for the calculation of the measure of a larger collection of subsets. This is often used in probability theory to calculate the probability of an event occurring.

Can a sigma algebra be infinite?

Yes, a sigma algebra can be infinite. In fact, many commonly used sigma algebras, such as the Borel sigma algebra, are infinite. This allows for the consideration of uncountably infinite sets in measure theory and probability theory.

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