Probability Spaces: Showing G is Field and not Sigma Field

[elliotician]

Homework Statement

Let $$\Omega = [0,1)$$

Let G be the collection of all subsets of $$\Omega$$ of the form
[a₁,b₁),$$\cup$$[a₂,b₂),$$\cup$$...$$\cup$$[a_r,b_r)
For r any non-negative integer and 0<=a₁
and a₁ <=b₁ <= a₂ ...

Show that G is a field

Show that G is not a $$\sigma$$-field

Homework Equations

Definition:
Let $$\Omega$$ be any set. A collection F of subsets of $$\Omega$$
is a field if:

1. $$\emptyset$$$$\in$$F
2. given A in F, then A^c=$$\Omega$$\A
3. given A and B in F, A$$\cup$$B is in F

In addition, the collection F is a $$\sigma$$-field if

given A₁, A₂, A₃... are all in F, so is there union

$$\bigcup$$A_i

The Attempt at a Solution

To show G is a field:

1.
Assume empty set is not in G, then there must be an element in empty which is not in the G - a contradiction since there are no elements in the empty set.

2.
If we have A in G then A=[a_i,b_i)
So A^c= $$\Omega$$/A

3. This is where I have the problem,
Set A in G as [a_i,b_i)
And B in G as [a_j,b_j)
So we want to show A U B in G
But if b_i= a_j
Then we will have A U B =[a_i,b_j)
Which is not in G

To show G is not a sigma field:
(0,1)= the union of intervals [1/n,1)
[1/n,1) is in G for all n
(0,1) is not in G
So G is not a sigma field.

So essentially the problem is with part 3 of the definition.

 

[EnumaElish]

Can you say "If we have A in G then A=[ai,bi)"? Isn't each element of G a union of r intervals? What would A look like when r > 1? Same comment applies to both part 2 and part 3 of your answer.

 

[elliotician]

Can you say "If we have A in G then A=[ai,bi)"? Isn't each element of G a union of r intervals? What would A look like when r > 1? Same comment applies to both part 2 and part 3 of your answer.

Thanks yes! OK so we say

A=[ai,bi)U[a(i+1),b(i+1)U...U[ar,br)

Ac = omega/A

Then 3 is now easy

A=[ai,bi)U[a(i+1),b(i+1)U...U[ar,br)
B=[aj,bj)U[a(j+1),b(j+1)U...U[as,bs)

so AUB = [ai,bi)U[a(i+1),b(i+1)U...U[ar,br)U[aj,bj)U[a(j+1),b(j+1)U...U[as,bs)

AUB=[ak,bk)U[a(k+1),b(k+1)U...U[at,bt)
where k=min{i,j} and t=max{r,s}

 

[EnumaElish]

What is the difference between 3 and the sigma-field property?

 

[elliotician]

For 3 we need the union of any two intervals to be in G
For the sigma field property we want all possible unions of all possible intervals to be in GPoint 3 basically requires [0,x) be in G for x<=1, which we do have.
In order for that sigma to hold we would need the interval (0,1) to be in the collection G.

Though I'm still not sure i fully understand the difference :S

 

[EnumaElish]

My understanding of probability fields is along these lines:

http://probability.infarom.ro/definition1.html

 

[elliotician]

OK well I am working with Probability and random processes By Geoffrey Grimmett, David Stirzaker, the google books preview covers the section i am dealing with (1.1 and 1.2 right at the start). This uses the definitions given in the exercise.

http://books.google.com/books?id=G3...rontcover&dq=probability+and+random+processes

In the link you have given i can't distinguish any definition of a field or sigma field, or any treatment of an infinite omega.