- #1

- 21

- 0

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

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

a) A={∅, {A,B,C}, {A}, {B}, {C}}

b) B={∅, {A,B,C}, {A}, {B,C}}

c) C={∅, {1,2,3}, {4,5}}

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

## 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: