Show that G is a sigma-algebra

[aaaa202]

Homework Statement

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

1) X\inG
2)A\inG => C(A)\inG
3)A_j \subset G => \bigcup A_j \in G

Show that G = {A\subsetX : #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\inG. But since G is only the set of proper subsets of X, i.e. doesn't contain X by definition, how can 1) hold?

 

[Mathitalian]

Maybe $\subset$ indicate "subset" instead of the symbol $\subseteq$...

 

[aaaa202]

wait what? No the definition clearly states to use proper subsets.

 

[vela]

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.

 

[aaaa202]

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.

 

[Office_Shredder]

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).

 

[Mathitalian]

[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) $A\cup (X\setminus A)= X\in G$

 

[Office_Shredder]

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"

 

[aaaa202]

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?

 

[Mathitalian]

Is N the cardinality of X right? Anyway, I'm quite sure that $\subset$ is used as $\subseteq$, so 1) holds... You have to check it on your book. If you want, take a look here

http://en.wikipedia.org/wiki/Subset#The_symbols_.E2.8A.82_and_.E2.8A.83