1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Probability Spaces

  1. Jun 18, 2009 #1
    1. The problem statement, all variables and given/known data
    Let [tex]\Omega = [0,1)[/tex]

    Let G be the collection of all subsets of [tex]\Omega[/tex] of the form
    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 [tex]\sigma[/tex]-field

    2. Relevant equations

    Let [tex]\Omega[/tex] be any set. A collection F of subsets of [tex]\Omega[/tex]
    is a field if:

    1. [tex]\emptyset[/tex][tex]\in[/tex]F
    2. given A in F, then Ac=[tex]\Omega[/tex]\A
    3. given A and B in F, A[tex]\cup[/tex]B is in F

    In addition, the collection F is a [tex]\sigma[/tex]-field if

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


    3. The attempt at a solution

    To show G is a field:

    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.

    If we have A in G then A=[ai,bi)
    So Ac= [tex]\Omega[/tex]/A

    3. This is where I have the problem,
    Set A in G as [ai,bi)
    And B in G as [aj,bj)
    So we want to show A U B in G
    But if bi= aj
    Then we will have A U B =[ai,bj)
    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.
    Last edited: Jun 18, 2009
  2. jcsd
  3. Jun 18, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    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.
  4. Jun 18, 2009 #3
    Thanks yes! OK so we say


    Ac = omega/A

    Then 3 is now easy


    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)

    where k=min{i,j} and t=max{r,s}
  5. Jun 18, 2009 #4


    User Avatar
    Science Advisor
    Homework Helper

    What is the difference between 3 and the sigma-field property?
  6. Jun 18, 2009 #5
    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 G

    Point 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
  7. Jun 18, 2009 #6


    User Avatar
    Science Advisor
    Homework Helper

  8. Jun 19, 2009 #7
    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.


    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.
    Last edited: Jun 19, 2009
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook