Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sigma field proofs

  1. Feb 4, 2012 #1
    I have the following to prove:

    Given a sigma field/sigma algebra B on a set S. Prove:

    i) 0 E B
    ii) if B1,..,Bk E B then UBi E B for i = 1 to n and nBi E B for i = 1 to n
    iii) if B1,B2... E B then nBi for i = 1 to infinity E B

    so this is what I have so far.

    i) A sigma algebra is defined as being non empty so therefore the 0 set should be in B at the very least.

    ii) I'm not sure how to prove this one. The union is in the third axiom of a sigma algebra. The union is defined as the collection of points that are in both sets, which should ultimately be in B. Should I use the power set here? Because if there are n elements in S then there are 2^n elements in B which is all possible combinations of S including S itself and therefore UBi and nBi should be contained in B.

    iii) this one just has the intersection from i to infinity. I think this should be a countably infinite set and from the power set, there would be an infinite number of combinations of S. So therefore the intersection should be contained in B?
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?