# How to construct a disjoint sequence from an infinite sigma-algebra?

1. Sep 18, 2010

### zzzhhh

Suppose $$\mathcal A$$ is an infinite $$\sigma$$-algebra, how to construct a disjoint sequence in $$\mathcal A$$ such that each term is nonempty? Thanks!

2. Sep 19, 2010

### zzzhhh

Last edited by a moderator: Apr 25, 2017
3. Jul 2, 2011

### stat22

Let X1, X2, X3, ..., Xn, ... are the elements of the said $\sigma$-algebra.

Then, for any positive integer n>1, let En = Xn - $[$Xn$\bigcap$$($Xn-1$\bigcup$ ... $\bigcup$X1$)$$]$

The result is a disjoint sequence {En}, where E1= X1

4. Jul 2, 2011

### micromass

And who says that the terms are nonempty?? For example, take X1=[0,2] and X2=[0,1]. Then

$$X_2\setminus (X_2\cap X_1)=[0,1]\setminus [0,1]=\emptyset$$

5. Jul 2, 2011

### stat22

micromass,

in your example X1={0,2} and X2={0,1}, the intersection is {0} not {0,1}. Thus X2-(X2$\bigcap$X1)={0,1}-{0}={1} not empty!

6. Jul 2, 2011

### micromass

I'm talking about the interval [0,1], not {0,1}.

But, if you don't like intervals, consider X1={0,1}, X2={0}. Then

$$X_2\setminus(X_2\cap X_1)=\emptyset$$

7. Jul 3, 2011

### Bacle

I think the standard way, given a collection A:={A_1,...,A_n,..} (I think this works only for countable collections ) to define B:={B_1,...,B_n,...} by:

B_1:=A_1
B_2:=A_2-A_1
......
......
B_n:=A_n-[A_1\/A_2\/......\/A_(n-1)]

8. Jul 3, 2011

### micromass

Yes, these are certainly disjoint, but perhaps empty!

9. Jul 3, 2011

### Bacle

Then I imagine one can ignore the empty sets, but that depends on what zzzhhh wants.

10. Jul 3, 2011

### micromass

Yes, but you still need to end up with an infinite number of sets. If you throw away too many sets, then you might end up with a finite number of sets...
The text passage is a discussion thread from a forum, where a user asks a question about constructing a disjoint sequence from an infinite sigma-algebra, and other users provide potential solutions and counterexamples.
single_question_with_answer

the passage contains a single mathematical question, and multiple answers or suggestions are provided by other users. however, the question is not fully resolved, as the provided answers are critiqued, and the final solution is not confirmed. nonetheless, answers are given, so it qualifies as a question with an answer.
Given a text passage, determine:
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First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
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system
# How to construct a disjoint sequence from an infinite sigma-algebra?

1. Sep 18, 2010

### zzzhhh

Suppose $$\mathcal A$$ is an infinite $$\sigma$$-algebra, how to construct a disjoint sequence in $$\mathcal A$$ such that each term is nonempty? Thanks!

2. Sep 19, 2010

### zzzhhh

Last edited by a moderator: Apr 25, 2017
3. Jul 2, 2011

### stat22

Let X1, X2, X3, ..., Xn, ... are the elements of the said $\sigma$-algebra.

Then, for any positive integer n>1, let En = Xn - $[$Xn$\bigcap$$($Xn-1$\bigcup$ ... $\bigcup$X1$)$$]$

The result is a disjoint sequence {En}, where E1= X1

4. Jul 2, 2011

### micromass

And who says that the terms are nonempty?? For example, take X1=[0,2] and X2=[0,1]. Then

$$X_2\setminus (X_2\cap X_1)=[0,1]\setminus [0,1]=\emptyset$$

5. Jul 2, 2011

### stat22

micromass,

in your example X1={0,2} and X2={0,1}, the intersection is {0} not {0,1}. Thus X2-(X2$\bigcap$X1)={0,1}-{0}={1} not empty!

6. Jul 2, 2011

### micromass

I'm talking about the interval [0,1], not {0,1}.

But, if you don't like intervals, consider X1={0,1}, X2={0}. Then

$$X_2\setminus(X_2\cap X_1)=\emptyset$$

7. Jul 3, 2011

### Bacle

I think the standard way, given a collection A:={A_1,...,A_n,..} (I think this works only for countable collections ) to define B:={B_1,...,B_n,...} by:

B_1:=A_1
B_2:=A_2-A_1
......
......
B_n:=A_n-[A_1\/A_2\/......\/A_(n-1)]

8. Jul 3, 2011

### micromass

Yes, these are certainly disjoint, but perhaps empty!

9. Jul 3, 2011

### Bacle

Then I imagine one can ignore the empty sets, but that depends on what zzzhhh wants.

10. Jul 3, 2011

### micromass

Yes, but you still need to end up with an infinite number of sets. If you throw away too many sets, then you might end up with a finite number of sets...
user
The text passage is a discussion thread from a forum, where a user asks a question about constructing a disjoint sequence from an infinite sigma-algebra, and other users provide potential solutions and counterexamples.

TYPE: SINGLE_QUESTION_WITH_ANSWER

The passage contains a single mathematical question, and multiple answers or suggestions are provided by other users. However, the question is not fully resolved, as the provided answers are critiqued, and the final solution is not confirmed. Nonetheless, answers are given, so it qualifies as a question with an answer.