Is the Intersection of a Sequence in a Sigma-Algebra Also in the Sigma-Algebra?

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In summary, to show that the intersection of a sequence of elements in a sigma-algebra F is also in F, we can use the property that the intersection can be written as the union of the complements, which are also in F. Therefore, the intersection must also be in F.
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Zaare
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Given the definition of a sigma-algebra, I need to show that the intersection of a sequence of elements in a sigma-algebra is in the sigma-algebra:

Given:
Let F be a sigma-algebra, then:
1) The empty set is in F.
2) If A is in F, then so is the complement of A.
3) The union of a sequence of elements in F is also in F.

To prove:
The intersection of a sequnce of elements in F is also in F.

I'm quite stuck and seem to go around in circles. Any help on how to attack this problem would be appreciated.
 
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  • #2
Take an intersection AnBnC... what is its complement?
 
  • #3
Right, the intersection is the union of the complements. And since the complements, and therefore the union, are in F, the intersection must also be in F.
 

FAQ: Is the Intersection of a Sequence in a Sigma-Algebra Also in the Sigma-Algebra?

What is a sigma-algebra?

A sigma-algebra is a collection of subsets of a given set that satisfies certain properties. It is used in measure theory to define the concept of measure.

What are the properties of a sigma-algebra?

A sigma-algebra must contain the empty set and the entire set, and must be closed under countable unions and complements.

How is a sigma-algebra related to measure theory?

A sigma-algebra is used to define the concept of measure, which is a mathematical way of assigning a numerical value to subsets of a given set.

Why is a sigma-algebra important in probability theory?

In probability theory, a sigma-algebra is used to define the events that can occur in a given sample space. It allows for the calculation of probabilities and the formulation of theorems and laws.

How do you construct a sigma-algebra?

A sigma-algebra can be constructed by starting with a collection of subsets of a given set and applying the properties of closure under countable unions and complements to generate the entire sigma-algebra.

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