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

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SUMMARY

This discussion focuses on proving that all elements of a sigma-algebra can be represented as unions of atoms, where the intersection of these atoms is an empty set. The participants clarify that if a sigma-algebra contains a finite number of elements, it is possible to identify elements \( A_i \) such that \( A_i \cap A_j = \emptyset \). The key insight is that atoms of the sigma-algebra, defined as minimal elements where only the empty set is contained, can be used to demonstrate that these atoms generate the entire sigma-algebra.

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  • Understanding of sigma-algebra and its properties
  • Familiarity with set theory concepts, particularly unions and intersections
  • Knowledge of the definition and significance of atoms in a sigma-algebra
  • Basic mathematical proof techniques
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  • Learn about the concept of atoms in measure theory
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Mathematicians, students of advanced mathematics, and anyone studying measure theory or set theory will benefit from this discussion.

somebd
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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!
 
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What are the elements?
 
somebd said:
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...
 
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.
 
somebd said:
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...
 
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...
 
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...
 

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