Proving Sigma-Rings Are Closed under Countable Intersections

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In summary, the conversation discusses trying to prove a concept using sigma-rings and symmetric difference. They consider using relative complements and X as a union of A1 and A_n. However, there is uncertainty about how this will lead to a solution.
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jdinatale
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I'm trying to prove the following and all I've got is like one line worth of proof.

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If we had that sigma-rings were closed under complementation, this would be easier, but we only know that if A in R and B in R, then A \ B in R and B \ A in R (symmetric difference). Is there a way to approach this using the symmetric difference?
 
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Instead of complementing take relative complements in A1.
 
  • #3


Office_Shredder said:
Instead of complementing take relative complements in A1.

Ok. So assume that [itex]A_1, A_2, A_3 ... \in R[/itex]. Then [itex]A_1 \backslash A_1, A_1 \backslash A_2, A_1 \backslash A_3 ... \in R[/itex]. Since [itex]R[/itex] is a [itex]\sigma[/itex]-ring, [itex]X = \cup_{n = 1}^\infty A_1 \backslash A_n\in R[/itex]. Also [itex]X \backslash A_1 \in R[/itex].

I'm not seeing where this is leading.
 

1. What is a sigma-ring?

A sigma-ring is a type of mathematical structure used in measure theory. It is a collection of sets that satisfies certain properties, including closure under countable unions and complements.

2. What does it mean for a sigma-ring to be closed under countable intersections?

This means that if you take any countable collection of sets from the sigma-ring and intersect them, the resulting set will also be in the sigma-ring. In other words, the sigma-ring is closed under taking finite intersections of its elements.

3. Why is proving that sigma-rings are closed under countable intersections important?

Proving this property is important because it is a key step in establishing the mathematical foundation for measure theory. It allows us to define measures on sigma-rings and use them to make precise calculations and predictions in various fields such as statistics, probability, and economics.

4. How is the proof for this property typically approached?

The proof for this property often involves using the definition of a sigma-ring and properties of set operations, such as De Morgan's laws and distributivity. It may also involve constructing a sequence of sets that converges to the desired intersection and showing that each set in the sequence is also in the sigma-ring.

5. Are there any real-life applications of this property?

Yes, there are many real-life applications of this property. For example, in statistics, this property is used to prove the existence of probability measures on sigma-rings, which are essential for making accurate predictions and decisions based on data. It is also used in economics to define and measure the size of sets of possible outcomes, such as the set of all possible price levels for a particular product.

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