Infinite union of sigma algebras

In summary, the sequence ##(B_n)_{n\in \mathbb{N}^{\ge 2}}## fails to be a subset of the set of all subsets of ##A_n##.
  • #1
fishturtle1
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Homework Statement
Let ##(X, \mathcal{A})## be a measurable space and ##(A_n)_{n\in\mathbb{N}}## be a strictly increasing sequence of ##\sigma## algebras. Show $$\mathcal{A}_\infty := \bigcup_{n\in\mathbb{N}} A_n$$ is never a ##\sigma## algebra.
Relevant Equations
A sigma algebra on ##X## is a subset of ##\mathcal{P}(X)## that contains the identity, is closed under complements, and closed under countable union.
For all ##n\in\mathbb{N}## we have ##\emptyset \in A_n##. Hence, ##\emptyset \in \mathcal{A}_\infty##. Let ##A \in \mathcal{A}_\infty##. Then ##A \in A_k## for some ##k\in\mathbb{N}##. So ##A^c \in A_k##. Hence, ##A^c \in \mathcal{A}_\infty##. Thus, ##\mathcal{A}_\infty## is closed under complements. So ##\mathcal{A}_\infty## must fail countable union.

We have ##A_1 \subsetneq A_2 \subsetneq A_3 \subsetneq \dots##. Let us define a sequence ##(B_n)_{n\in \mathbb{N}^{\ge 2}}## where ##B_n## is some set in ##A_n## but not in ##A_{n-1}##.

Also, if ##\mathcal{A}_\infty## is closed under countable union, then ##\mathcal{A}_\infty## is closed under countable intersection. So maybe that's how to get a contradiction?

Consider ##\bigcap_{n\ge 2} B_n##. Is this on the right track?...
 
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  • #3
Math_QED said:
This is a difficult problem. You will need to work a lot harder than your attempt shows.

Here is a solution: https://math.stackexchange.com/ques...igma-algebras-is-not-a-sigm?noredirect=1&lq=1

Obviously don't click it if you don't want to get spoiled. If you want to, I can give you some hints.

Thanks for the reply. I tried working with the sequence ##(B_n)## where ##B_n## is an element of ##\mathcal{A}_{n+1} \setminus \mathcal{A}_n##. We want to show ##\bigcup_{n} B_n \not\in \mathcal{A}_\infty##. We can observe that ##B_1 \in \mathcal{A}_2\setminus\mathcal{A}_1, B_2 \in \mathcal{A}_3\setminus\mathcal{A_2} \supset \mathcal{A}_3 \setminus \mathcal{A}_1 \dots## and so for all ##n## we have ##B_n \not\in \mathcal{A}_1##. Similarly, for all ##n \ge 2## we have ##B_n \not\in \mathcal{A}_2## and continuing in this way, for all ##n \ge k## we have ##B_n \not\in \mathcal{A}_k##.

Assume by contradiction that ##\bigcup_n B_n \in \mathcal{A}_m## for some ##m##. Then, ##B_1, B_2, \dots, B_{m-1} \in \mathcal{A}_m##. Since ##\mathcal{A}_m## is closed under complements and countable intersection, we have ##\bigcup_{k=m}^{\infty}B_k \in \mathcal{A}_m##. By construction, ##B_m \not\in \mathcal{A_m}##. Under these assumptions, can we show ##B_m \in \mathcal{A}_m## to get a contradiction?

Thanks for the stack exchange link (I haven't clicked on it yet but maybe if this problem turns out too be too hard I will...) its funny, my homework was too hard so I found this problem in textbook and thought it'd be be a fun one to do as a warm up... if you have time, would you be able to give me a hint, please?
 
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  • #4
It's definitely no warm-up problem haha. It seems to be a statement that was published (together with its proof) in

A. Broughton and B. W. Huff: A comment on unions of sigma-fields. The American Mathematical Monthly, 84, no. 7 (1977), 553-554

I don't think the approach you suggest works. It was my first idea too when I saw the question this morning but I don't think you can make it work like that, as you don't know how the sets relate.

I underestimated the problem. I don't think I can give a "good" hint that will make you solve the problem, as the solution in the link gets quite technical, so I suggest reading the answer in the link. I think it will already be a good exercise to make sure you understand that answer! This is the kind of question on which you spend several days to solve.

Now, I'm curious. What were the "hard" questions in your measure theory exercise session?
 
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  • #5
Math_QED said:
It's definitely no warm-up problem haha. It seems to be a statement that was published (together with its proof) in

A. Broughton and B. W. Huff: A comment on unions of sigma-fields. The American Mathematical Monthly, 84, no. 7 (1977), 553-554

I don't think the approach you suggest works. It was my first idea too when I saw the question this morning but I don't think you can make it work like that, as you don't know how the sets relate.

I underestimated the problem. I don't think I can give a "good" hint that will make you solve the problem, as the solution in the link gets quite technical, so I suggest reading the answer in the link. I think it will already be a good exercise to make sure you understand that answer! This is the kind of question on which you spend several days to solve.

Now, I'm curious. What were the "hard" questions in your measure theory exercise session?

Wow, ok! I'll do as you suggested then and go through the proof.

As to the hard questions i'd prefer not to say, since they're homework but it's on L^p spaces as well as Construction of measures. Sorry I know that's not a satisfying answer =\...

Thank you for your time on this.
 
  • #6
fishturtle1 said:
Wow, ok! I'll do as you suggested then and go through the proof.

As to the hard questions i'd prefer not to say, since they're homework but it's on L^p spaces as well as Construction of measures. Sorry I know that's not a satisfying answer =\...

Thank you for your time on this.

I understand. Have a good day!
 

1. What is an infinite union of sigma algebras?

An infinite union of sigma algebras is a collection of sigma algebras that contains all possible subsets of a given set, including an infinite number of subsets. This means that the union includes all possible combinations of elements from the set, making it a very large and comprehensive collection.

2. How is an infinite union of sigma algebras different from a finite union?

An infinite union of sigma algebras differs from a finite union in that it includes an infinite number of subsets, while a finite union only includes a limited number of subsets. This makes the infinite union more comprehensive and allows for a wider range of possibilities to be included in the collection.

3. What is the purpose of an infinite union of sigma algebras?

The purpose of an infinite union of sigma algebras is to create a comprehensive collection of subsets that can be used to analyze and understand a given set. This allows for a more thorough and detailed examination of the set, which can be useful in various scientific fields such as statistics, probability, and measure theory.

4. How is an infinite union of sigma algebras used in scientific research?

An infinite union of sigma algebras is used in scientific research to analyze and understand complex systems or sets. It allows for a more comprehensive and detailed examination of the data, which can lead to new insights and discoveries. This is particularly useful in fields such as mathematics, physics, and computer science.

5. Are there any limitations to using an infinite union of sigma algebras?

While an infinite union of sigma algebras can provide a comprehensive collection of subsets, it may not always be practical or necessary to use in scientific research. In some cases, a finite union may be sufficient for the analysis and understanding of a set. Additionally, the size and complexity of an infinite union can make it challenging to work with and may require specialized techniques and tools.

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