Proving the Representation of Sigma-Algebra Elements as Unions of Atoms

[somebd]

Hello!
How can you proof that all elements of sigma-algebra can be represented as unions of the elements intersection of which is an empty set? I am out of ideas :( Your help would be appreciated!

 

[mathman]

What are the elements?

 

[micromass]

Hello!
How can you proof that all elements of sigma-algebra can be represented as unions of the elements intersection of which is an empty set? I am out of ideas :( Your help would be appreciated!

I'm afraid that I don't really understand the question... Can you give the exact wording of the question, or perhaps some extra information. Or perhaps write it in symbols could help...

 

[somebd]

Sorry I can't type symbols since I am on my phone :( thanks for your interest!
Basically, the task (which is formulated in language other than English so I'm sorry for possible mistakes) is to show that if sigma-algebra has a finite number of elements, there are elements Ai, i=1,...,n for which Ai intersection with Aj (i and j are indices) is an empty set and that every sigma-algebra element can be represented as unions of these sets Ai. I have shown the first part (about intersection) but stuck with the second.

 

[micromass]

Sorry I can't type symbols since I am on my phone :( thanks for your interest!
Basically, the task (which is formulated in language other than English so I'm sorry for possible mistakes) is to show that if sigma-algebra has a finite number of elements, there are elements Ai, i=1,...,n for which Ai intersection with Aj (i and j are indices) is an empty set and that every sigma-algebra element can be represented as unions of these sets Ai. I have shown the first part (about intersection) but stuck with the second.

Ah, I understand! So you said you have shown the first part. So you found elements Ai for which $$A_i\cap A_j=\emptyset$$. So, which elements did you find. We'll see if those elements also satisfy the second condition...

 

[somebd]

Oh well, I am starting to doubt now :D But I just thought that if A and B belong to sigma-alegebra, then so do sets A\B and B\A and their intersection is an empty set...

 

[micromass]

All right, certainly A/B and B/A satisfy the first condition (their intersection being empty), but they don't satisfy the other one (the unions of such elements generating the sigma-algebra).

The trick is looking at the so-called atoms of the algebra. Let $$\mathcal{B}$$ be your sigma-algebra, then A is called an atom of $$\mathcal{B}$$ if

$$\forall B\in \mathcal{B}:~B\subseteq A~\Rightarrow~B=\emptyset~\text{or}~B=A$$.

Thus the atoms are these elements of $$\mathcal{B}$$ such that only the empty set is contained in them. So in some sense, they are the minimal elements of the sigma-algebra.
Now let $$\mathcal{A}$$ be the set of all the atoms. Try to prove that this set satisfies your two conditions: the intersection is zero, and they generate the algebra...