Show that G is a sigma-algebra

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In summary: I think it's better to use ##\subseteq## or ##\subsetneq## to avoid ambiguities, but sometimes people use ##\subset## as ##\subseteq##.In summary, a σ-algebra on a set X is a family of subsets of X that includes X itself and satisfies three properties: (1) X is in the family, (2) the complement of any subset in the family is also in the family, and (3) the union of any number of subsets in the family is also in the family. In order to show that G = {A⊂X : #A≤N or ≠C(A)≤N} is a sigma-algebra on X, it
  • #1
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Homework Statement


A σ-algebra G on a set X is a family of subsets of X satisfying:

1) X[itex]\in[/itex]G
2)A[itex]\in[/itex]G => C(A)[itex]\in[/itex]G
3)Aj [itex]\subset[/itex] G => [itex]\bigcup[/itex] Aj [itex]\in[/itex] G

Show that G = {A[itex]\subset[/itex]X : #A≤N or ≠C(A)≤N}

# stands for the cardinality of the set.

Homework Equations





The Attempt at a Solution


Actually I am not so far in the problem solving because I am stuck at showing the first property. We must have that X[itex]\in[/itex]G. But since G is only the set of proper subsets of X, i.e. doesn't contain X by definition, how can 1) hold?
 
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  • #2
Maybe ##\subset## indicate "subset" instead of the symbol ##\subseteq##...
 
  • #3
wait what? No the definition clearly states to use proper subsets.
 
  • #4
Why are you trying to show the first property holds? Isn't it being given to you as part of a definition? In other words, you can assume G satisfies those three properties in trying to write your proof.
 
  • #5
I see I made a mistake. I meant to write: Show that G = {A⊂X : #A≤N or ≠C(A)≤N} is a sigma-algebra on X.
 
  • #6
The proper subset symbol is often used to denote improper subset so I wouldn't get too caught up in the details of what they are trying to say there.

However the typical definition of a sigma-algebra does not say that X is in G, it just says that there exists some subset A of X which is contained in G (i.e. G is not empty).
 
  • #7
Office_Shredder said:
[omissis]
However the typical definition of a sigma-algebra does not say that X is in G, it just says that there exists some subset A of X which is contained in G (i.e. G is not empty).

Yes, you're right, but if ##A\in G\implies X\setminus A\in G## so, by 3) ##A\cup (X\setminus A)= X\in G ##
 
  • #8
Oops that's embarassing. Then I retract that point and am sticking with "nobody uses the proper subset symbol and means it unless they explicitly state so, so X is contained in G"
 
  • #9
but what if we can't take a proper subset of G? Shouldn't we allow for the case where you have to take all of G (i.e. an improper subset) if we want the argument in #7 to hold?
 
  • #10

FAQ: Show that G is a sigma-algebra

What is a sigma-algebra?

A sigma-algebra is a collection of subsets of a given set that satisfies certain properties, such as containing the empty set, being closed under complementation and countable unions.

How is a sigma-algebra different from a regular algebra?

A sigma-algebra is a more general concept than a regular algebra, as it allows for countable unions in addition to finite unions. This makes it more useful for dealing with infinite sets.

Why is it important to show that G is a sigma-algebra?

It is important to show that G is a sigma-algebra because it proves that G has the necessary properties for it to be used as a measure space in mathematical analysis and probability theory.

How can I prove that G is a sigma-algebra?

To prove that G is a sigma-algebra, you need to show that it contains the empty set, is closed under complementation, and is closed under countable unions. This can be done by using logical arguments and mathematical proofs.

What are some real-world applications of sigma-algebras?

Sigma-algebras are used in various fields such as statistics, economics, and finance to model and analyze complex systems that involve infinite or uncountable outcomes. They are also essential in probability theory for defining and calculating probabilities of events.

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