Quick question about sigma algebra proof.

In summary, a sigma algebra is a mathematical structure that is used to define a probability measure on a set. It is a collection of subsets that contains the empty set, is closed under complementation, and is closed under countable unions. This structure is commonly used in probability theory and measure theory. The terms "sigma algebra" and "sigma field" are interchangeable. To prove that a collection of subsets is a sigma algebra, one must show that it satisfies the three properties mentioned above. A sigma algebra can contain an uncountable number of sets as long as it satisfies these properties.
  • #1
Kuma
134
0
If A is a sigma algebra on a set S.

if Ai E A. i = 1,...,n
prove that the intersection of all Ai E A.

now isn't the intersection just the empty set in this case? Which is already proved as the empty set is always contained in A?
 
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  • #2
Why would the intersection of the Ai be empty?
 

FAQ: Quick question about sigma algebra proof.

1. What is a sigma algebra?

A sigma algebra is a collection of subsets of a given set that satisfies three properties: it contains the empty set, it is closed under complementation, and it is closed under countable unions. This mathematical structure is commonly used in probability theory and measure theory.

2. What is the purpose of a sigma algebra?

A sigma algebra is used as a way to define a probability measure on a set. By specifying a sigma algebra, we are able to determine which events are considered "measurable" and thus assign probabilities to them.

3. What is the difference between a sigma algebra and a sigma field?

There is no difference between a sigma algebra and a sigma field. These terms are used interchangeably to refer to the same mathematical structure.

4. How do you prove that a collection of subsets is a sigma algebra?

To prove that a collection of subsets is a sigma algebra, you must show that it satisfies the three properties: it contains the empty set, it is closed under complementation, and it is closed under countable unions. This can be done by explicitly checking each property for each subset in the collection.

5. Can a sigma algebra contain an uncountable number of sets?

Yes, a sigma algebra can contain an uncountable number of sets. As long as the collection satisfies the three properties, it can contain any number of sets, countable or uncountable.

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