Identifying Potential σ-Fields: Exam Practice Exercises

[Charlotte87]

I have started to solve exercises given on a previous exam, but typically I do not have the answers.

Homework Statement

The question is: Which (if any) of these collections are potentially σ-fields over some sample space such that probability functions could be defined over them? Explain briefly

Homework Equations

a) A={∅, {A,B,C}, {A}, {B}, {C}}
b) B={∅, {A,B,C}, {A}, {B,C}}
c) C={∅, {1,2,3}, {4,5}}

The Attempt at a Solution

I started by setting up the conditions for a these sets to be σ-fields. Let B be a collection of subsets of ℂ, then B is a σ-field is:
1) The empty set is part of the subset
2) If x \in B, then x^{C}\in B
3) If the sequence of sets {X_1, X_2, X_3,...} is in B, then \bigcupX_i is a part of ℂ

The first condition is satisfied for all the collections.

a) If I have understood the theory of complements right we have that:
∅^{C}={A,B,C} - OK
{(A,B,C)}^{C}=∅ - OK
{A}^{C}={B,C} - not in the collection -> not a σ-field

b) In the same manner, I find that all the complements exist in this collection. For the third part i have writte \bigcup{(A,B,C)}\cup{(A)}\cup{(B,C)}=∅. Will that be correct?
Anyway, I conclude that this is a potentially σ-field.

c) Here I find that the complement of the empty set does not exist, and thus this is not a σ
-field

 

[micromass]

For (b), are you sure that every complement is again in the set??

 

[Charlotte87]

Well, I believe so.
The complement of the empty set is {A,B,C}.
The complement of {A,B,C} is the empty set.
The complement of {A} is {B,C}
The complement of {B,C} is {A},

Am I missing something?

 

[micromass]

Well, I believe so.
The complement of the empty set is {A,B,C}.
The complement of {A,B,C} is the empty set.
The complement of {A} is {B,C}
The complement of {B,C} is {A},

Am I missing something?

I don't see {B,C} in the set...

 

[Charlotte87]

Sorry, my mistake, it should be {B,C}, not {A,B}.

 

[micromass]

OK, then you're right that (b) is sigma-field. However, your explanation

\bigcup{(A,B,C)}\cup{(A)}\cup{(B,C)}=∅.

does not seem right. It's a bit weird that the union of all these things could be empty.

 

[Charlotte87]

Your right, I have been confusing it with the intersections. So, the union of all these things will be {A,B,C}, right? and that should of course be a part of the larger set.

 

[micromass]

Yes, but that's only the union of all the sets. That is, you then know that

\emptyset\cup \{A,B,C\}\cup \{A\}\cup \{B,C\}

is in the sigma-field.

But you also need to check the other unions, like

\{A\}\cup\{B,C\}

and

\{A\}\cup\{A,B,C\}

and so on.

Of course, these will lie trivially in the sigma-field again, but you need to check it.

 

[Charlotte87]

Thankyou very much. This has been very helpful!