Set thory: sigma-fields

  • #1
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 [itex]\in[/itex] B, then [itex]x^{C}[/itex][itex]\in[/itex] B
3) If the sequence of sets {X_1, X_2, X_3,...} is in B, then [itex]\bigcup[/itex]X_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:
[itex]∅^{C}[/itex]={A,B,C} - OK
[itex]{(A,B,C)}^{C}[/itex]=∅ - OK
[itex]{A}^{C}[/itex]={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 [itex]\bigcup{(A,B,C)}\cup{(A)}\cup{(B,C)}=∅[/itex]. 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
 
Last edited:

Answers and Replies

  • #2
22,089
3,296
For (b), are you sure that every complement is again in the set??
 
  • #3
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?
 
  • #4
22,089
3,296
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...
 
  • #5
Sorry, my mistake, it should be {B,C}, not {A,B}.
 
  • #6
22,089
3,296
OK, then you're right that (b) is sigma-field. However, your explanation

[itex]\bigcup{(A,B,C)}\cup{(A)}\cup{(B,C)}=∅[/itex].

does not seem right. It's a bit weird that the union of all these things could be empty.
 
  • #7
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.
 
  • #8
22,089
3,296
Yes, but that's only the union of all the sets. That is, you then know that

[tex]\emptyset\cup \{A,B,C\}\cup \{A\}\cup \{B,C\}[/tex]

is in the sigma-field.

But you also need to check the other unions, like

[tex]\{A\}\cup\{B,C\}[/tex]

and

[tex]\{A\}\cup\{A,B,C\}[/tex]

and so on.

Of course, these will lie trivially in the sigma-field again, but you need to check it.
 
  • #9
Thankyou very much. This has been very helpful!
 

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