Infinite union of sigma algebras

[fishturtle1]

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?...

 

[member 587159]

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.

 

[fishturtle1]

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?

 

[member 587159]

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?

 

[fishturtle1]

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.

 

[member 587159]

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!