What is the issue with the example of \sigma-algebras on infinite sets?

In summary, the conversation discusses the definitions of an algebra and a \sigma-algebra on an arbitrary set, as well as an example that shows a collection not being closed under the formation of countable unions. It also includes a question about finding an infinite subset whose complement is also infinite. The summary concludes with a summary of the logic used to determine whether a collection is a \sigma-algebra or not.
  • #1
tavrion
9
0
I am trying to better my understanding of [itex]\sigma[/itex]-[itex]algebra[/itex]s, and I have a bit of an issue with one of the examples. This is from Cohn Measure Theory, and before I give the problem, here are two definitions:

Let [itex]X[/itex] be an arbitrary set. A collection [itex]\delta\Large[/itex] of subsets of [itex]X[/itex] is an [itex]algebra[/itex] on [itex]X[/itex] if:

(a) [itex]X \in Z[/itex]

(b) for each set [itex]A[/itex] that belongs to [itex]\delta\Large[/itex], the set [itex]A^c[/itex] belongs to [itex]\delta\Large[/itex]

(c) for each finite sequence [itex]A_{1}, ... , A_{n}[/itex] of sets that belong to [itex]\delta\Large[/itex] the set [itex]\bigcup_{i=1}^{n}A_{i}[/itex] belongs to [itex]\delta\Large[/itex] and

(d) for each finite sequence [itex]A_{1}, ... , A_{n}[/itex] of sets that belong to [itex]\delta\Large[/itex] the set [itex]\bigcap_{i=1}^{n}A_{i}[/itex] belongs to [itex]\delta\Large[/itex]



Let [itex]X[/itex] be an arbitrary set. A collection [itex]\delta\Large[/itex] of subsets of [itex]X[/itex] is a [itex]\sigma[/itex]-[itex]algebra[/itex] on [itex]X[/itex] if:

(a) [itex]X \in Z[/itex]

(b) for each set [itex]A[/itex] that belongs to [itex]\delta\Large[/itex], the set [itex]A^c[/itex] belongs to [itex]\delta\Large[/itex]

(c) for each infinite sequence [itex]\{A_{i}\}[/itex] of sets that belong to [itex]\delta\Large[/itex] the set [itex]\bigcup_{i=1}^{\infty}A_{i}[/itex] belongs to [itex]\delta\Large[/itex] and

(d) for each infinite sequence [itex]\{A_{1}\}[/itex] of sets that belong to [itex]\delta\Large[/itex] the set [itex]\bigcap_{i=1}^{\infty}A_{i}[/itex] belongs to [itex]\delta\Large[/itex]


Okay, with all that. Here is what I am having issues with.

If [itex]X[/itex] is an infinite set, and [itex]\delta[/itex] is the collection of all subsets [itex]A[/itex] such that either [itex]A[/itex] or [itex]A^c[/itex] is finite. Then [itex]\delta[/itex] is an algebra on [itex]X[/itex] but not closed under the formation of countable unions, and so not a [itex]\sigma[/itex]-[itex]algebra[/itex].

So, if I take, for example [itex]X[/itex] to be the set of all positive integers, that is [itex]X = {1,2,3,...}[/itex] and define [itex] A_{i} = i [/itex].

Then, I have [itex]\bigcup_{i=1}^{n}A_{i} = {1,2,3,...,n}[/itex] which belongs to [itex]\delta[/itex] but [itex]\bigcup_{i=1}^{\infty}A_{i} = {1,2,3,...}[/itex] belongs to [itex]\delta[/itex] as well, so why does this fail to be a [itex]\sigma[/itex]-[itex]algebra[/itex]? Where have I gone wrong?
 
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  • #2
Can you think of an infinite subset of X whose complement is also infinite?
 
  • #3
Thank you. Let me see if I understand this right,

As before with [itex] X = {1,2,3,...} [/itex] and

if [itex]A_{i} = 2i[/itex] and [itex]A_{j} = 2j-1[/itex], then [itex]A = \bigcup_{i}^{\infty}A_{i} = 2,4,6,... [/itex] and [itex]A_{\infty}^c = \bigcup_{j}^{\infty}A_{j} = 1,3,5,... [/itex]

If [itex]A^c[/itex] is defined to be finite, we have [itex]A_{n}^c = \bigcup_{j}^{n}A_{j} = 1,3,5,...,2n-1 [/itex].

So, we have [itex] X \notin A_{n}^c\cup{A}[/itex] which fails to be a [itex]\sigma[/itex]-[itex]algebra[/itex] but [itex]X \in A_{\infty}^c\cup{A}[/itex] is a [itex]\sigma[/itex]-[itex]algebra[/itex]?

Is this logic correct?
 

Related to What is the issue with the example of \sigma-algebras on infinite sets?

1. What is a Sigma Algebra on an infinite set?

A Sigma Algebra on an infinite set is a collection of subsets of the set that satisfies certain properties. Specifically, it is a collection of subsets that is closed under countable unions and complements. This means that if a set is in the Sigma Algebra, its complement and any countable unions of sets in the Sigma Algebra are also in the Sigma Algebra.

2. Why is a Sigma Algebra important in mathematics and science?

Sigma Algebras are important because they provide a way to define a measure on a set. A measure is a function that assigns a non-negative number to subsets of a set, and it is used to quantify the size or volume of a set. Sigma Algebras allow us to define measures on more complicated sets, such as infinite sets or sets with irregular shapes, which is useful in many areas of mathematics and science.

3. How is a Sigma Algebra different from a sigma-field?

A Sigma Algebra is a generalization of a sigma-field. A sigma-field is a collection of subsets of a set that is closed under countable unions, countable intersections, and complements. A Sigma Algebra only needs to be closed under countable unions and complements. Therefore, a sigma-field is a special type of Sigma Algebra.

4. Can a Sigma Algebra on an infinite set be countably infinite?

Yes, a Sigma Algebra on an infinite set can be countably infinite. In fact, the Sigma Algebra on an infinite set with countably infinite elements is the most common type of Sigma Algebra. This is because it is often easier to define a measure on a countably infinite set compared to an uncountable set.

5. How is a Sigma Algebra related to probability theory?

A Sigma Algebra is closely related to probability theory because it allows us to define a probability measure on a set. This probability measure assigns a number between 0 and 1 to subsets of the set, and it represents the likelihood or chance of that subset occurring. Sigma Algebras are used in probability theory to define events and calculate probabilities in a rigorous and consistent manner.

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