00:01
Okay, so we've got f1 and f2 are sigma fields on omega.
00:11
And what this means, the definition of a sigma field is that a omega lies within these fields.
00:34
If a set a lies in either of the fields, then so does the complement of a.
00:43
And finally if a1, a2, an infinite set of sets lie within either f then so does their union and so we just need to prove that these properties hold for f1 intersect f2.
01:20
So firstly we have that since we know omega lies in f1 and omega lies in f2 that's what this says up here and therefore omega lies in both f1 and f2 it lies in the intersection as well so that's property satisfied then we say that if a is in f1 intersect f2 then we know that a is in f1 which by property b above implies that a is in f1 c and we also know that a is in f2 because because it's in the intersection, so it has to be an f2.
02:07
And therefore by the above, a is in f2c.
02:13
And then the fact that it's in both means that it's in the intersection of the two.
02:23
Sorry, i've gotten my cs the wrong way around.
02:25
It should be, property b tells us that ac lies in f1 and ac lies in f2, and therefore ac lies in the intersection of f1 and f2.
02:37
And then finally we've got that if a1, a2, so on lie in f1 and f2 then again they lie in f1 and they lie in f2 and so by property c above by the fact that f1 and f2 are sigma fields we know that the union lies in f1 and the union also lies in f2, and therefore the union lies in the intersection.
03:18
So that's true.
03:20
For question two, we're asked to prove that this is a probability measure.
03:32
And so for that, we need to show that it's always bigger than zero...