Why is a Dynkin system not a Pi-system

Thread starter tunaaa
Start date May 25, 2012
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In summary, a Dynkin system is a collection of sets that are closed under finite intersections and complements, commonly used in probability theory to define measures. A Pi-system, also known as a semi-ring, is a collection of sets that is closed under finite intersections and contains the empty set, commonly used in measure theory and probability theory. While both Dynkin systems and Pi-systems are closed under finite intersections, Dynkin systems also have the property of being closed under complements, making them a more general and flexible collection of sets compared to Pi-systems. This means that a Dynkin system is not a Pi-system, but a Pi-system can also be a Dynkin system as it is a subset of the larger collection of sets.

May 25, 2012

tunaaa

This appears to be the only difference between a sigma-algebra and a Dynkin system:

Sigma-algebra is closed under countable union
Dynkin system is closed under countable union of disjoint sets

This seems to result in the D-system not being a pi-system (while the sigma-alg is). Why?

Many thanks.

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May 28, 2012

camillio

Hint: think about complements.

1. What is a Dynkin system?

A Dynkin system is a collection of sets that satisfy certain properties, such as closure under finite intersections and complements. It is often used in probability theory to define measures.

2. What is a Pi-system?

A Pi-system, also known as a semi-ring, is a collection of sets that is closed under finite intersections and contains the empty set. It is commonly used in measure theory and probability theory.

3. What is the difference between a Dynkin system and a Pi-system?

While both Dynkin systems and Pi-systems are collections of sets that are closed under finite intersections, Dynkin systems also have the property of being closed under complements. This means that a Dynkin system is a larger collection of sets compared to a Pi-system.

4. Why is a Dynkin system not a Pi-system?

A Dynkin system is not a Pi-system because it has the additional property of closure under complements. This means that a Dynkin system contains more sets than a Pi-system, making it a more general and flexible collection of sets.

5. Can a Pi-system also be a Dynkin system?

Yes, a Pi-system can also be a Dynkin system. This is because a Pi-system is a subset of a Dynkin system, meaning that all of the properties of a Pi-system are also satisfied by a Dynkin system.