Algebras and Sigma-Algebras

  • Thread starter MathNerd
  • Start date
In summary, in a college level probability course, the difference between an algebra and a sigma-algebra on a fixed set S is that a sigma algebra is closed with respect to countable unions. We require an event space to be a sigma-algebra instead of an algebra because we want probability/measure to be sigma-additive, which allows for probability calculations involving limits of sequences. Additionally, we can find a set S and an algebra A on S such that A is not a sigma-algebra on S, specifically the system of finite and co-finite subsets of any infinite set. Finally, it can be proven that every sigma-algebra on S is also an algebra on S, using definitions of algebra and sigma-algebra and potentially De Morgan
  • #1
MathNerd
1
0
I'm currently taking a college level probability course and I am stuck on a couple questions involving algebras and sigma-algebras.

Let S be a fixed set.

1. What is the difference between an algebra on S and a sigma-algebra on S?

2. Why do we require an event space to be a sigma algebra instead of an algebra?

3. Find a set S and an algebra A on S such that A is not a sigma-algebra on S.


Also, I have a proof that I could use some hints on how to start and the general form in which I should go about it.

Prove that every sigma-algebra on S is an algebra on S.

Thanks.
 
Physics news on Phys.org
  • #2
I suspect it has something to do with sigma-algebras being limited to only finite intersections.

Continuous random variables cause problems of you assume countable additivity of probability for point events.

--Elucidus
 
  • #3
MathNerd said:
I'm currently taking a college level probability course and I am stuck on a couple questions involving algebras and sigma-algebras.

Let S be a fixed set.

1. What is the difference between an algebra on S and a sigma-algebra on S?
A sigma algebra is an algebra closed wrt countable unions.
2. Why do we require an event space to be a sigma algebra instead of an algebra?
Because we want probability/measure to be sigma-additive.
3. Find a set S and an algebra A on S such that A is not a sigma-algebra on S.
The system of finite and co-finite subsets of any infinite set forms an algebra which is not a sigma algebra.
Prove that every sigma-algebra on S is an algebra on S.
What definitions of algebra and sigma-algebra are you using? Under most definitions, this would be trivial (perhaps requiring the use of de Morgan's laws).
 
  • #4
Why do we require an event space to be a sigma algebra instead of an algebra?

Preno said:
Because we want probability/measure to be sigma-additive.

Perhaps not entirely responsive... the next question would be "why do we want that?"

So ... How about:

In order to do probability calculations involving limits of sequences. Either sequences of events or more generally sequences of random variables.
 

1. What is an algebra?

An algebra is a mathematical structure that consists of a set of elements and a collection of operations that can be performed on those elements. The operations typically include addition, subtraction, multiplication, and division, and they must follow certain rules to be considered an algebra.

2. What is a sigma-algebra?

A sigma-algebra is a type of algebra that is used in the field of measure theory. It is a collection of subsets of a given set that has certain properties, such as being closed under countable unions and complements. Sigma-algebras are important in probability theory and other areas of mathematics.

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

The main difference between an algebra and a sigma-algebra is that a sigma-algebra must satisfy the additional property of being closed under countable unions. This means that if you take any countable number of sets from the sigma-algebra and combine them with a union, the resulting set is still part of the sigma-algebra. Algebras do not necessarily have this property.

4. How are algebras and sigma-algebras used in probability theory?

Algebras and sigma-algebras are used in probability theory to define the concept of a measurable space, which is a set of events that can be assigned probabilities. The elements of the algebra or sigma-algebra are considered to be the measurable events, and the operations within the algebra dictate how probabilities are assigned to those events.

5. Can you give an example of an algebra and a sigma-algebra?

An example of an algebra would be the set of all subsets of the real numbers that can be formed by adding, subtracting, multiplying, or dividing any two real numbers. An example of a sigma-algebra would be the set of all subsets of the real numbers that can be formed by taking the countable union or complement of any set in the algebra.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
2K
  • Science and Math Textbooks
Replies
3
Views
261
  • Set Theory, Logic, Probability, Statistics
Replies
8
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
Back
Top