# Sigma field proofs

1. Feb 4, 2012

### Kuma

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?