[944] Question about Stoc-cal

[Gautam Iyer]

On Sat, Oct 28, 2017 at 06:42:27PM -0700, Li Wang wrote:

> Let g be a sigma algebra, are we sure that empty set and omega are
> always in such g. If it is not the case, can we always say P(empty
> set)=0 and P(omega) = 1? Does the probability measure depend on g? If
> empty set and omega are not in a certain g, can we still write P(empty
> set) and P(omega)?

Hi Li, and everyone else,

If you're asking math questions, please send it to the whole list and
not only to me. Odds are it may (or may not) help others, and someone
else may (or may not) be able to respond sooner. I will always direct
responses to math questions to the whole list.

To answer your question, yes. If G is non-empty, then you can check
directly from the two assumptions I gave you that both the empty set and
the full set Omega belong to G. Indeed, take any set A in G. Then by
assumption A complement also belongs to G. Hence so does A union A
complement. That means Omega belongs to G. Thus Omega complement (which
is the empty set) also belongs to G.

Warning -- if a set belongs to G, it DOES NOT MEAN that all subsets of
it belong to G. (See Davids response for more info, and I will likely
mention it in class later.)

Best,

Gautam

PS: Also David's scanned notes from the recitation are online.

-- 
Alternative definitions of terms from Math Lectures:
TWO LINE PROOF: The rest of the lecture will be spent writing fifty odd
lines, after which the proof will still be incomplete.