Basic stats question involving borel sets

[bennyska]

Homework Statement

http://i.imgur.com/tjpka.png (the actual problem is the third part down)

Homework Equations

the first two parts are the definition of borel sets,and the second part is a relevant theorem.

The Attempt at a Solution

so I'm new to Borel sets. And I feel like I'm missing something big, because this exercise seems to contradict a lot of statistics I've learned. I have a feeling once I get what I'm missing, it should be relatively easy to prove this stuff, but if someone could help me find out what it is I'm missing, it would be greatly appreciated.

For example: (i) from the exercise. This seems to be true if and only if A and B are disjoint, but nowhere are we told this is so. (ii) seems to imply that AUB = 1 (following from (iv) of the theorem), but again, we are not told this. (iii) seems to imply that A⊆C, or C⊆A, but nowhere are we told this is so. (iv) I haven't started yet, but I'm not worried about that one; at first glance it just looks like letting AUB equal one set, and then using the theorem. But the first 3 are proving a big conceptual block for me.

again, I think I'm missing exactly what a Borel set is, and how it is different than a usual set. Any help with that would be awesome. Thanks.

 

[LCKurtz]

Well, it's been about 45 years since I taught any stat courses and, believe it or not, even teachers forget eventually . But, for what it's worth, I don't think you are missing anything. It looks to me like some hypotheses are left out, as you suspect.

 

[voko]

(i) of 1.3.5 clearly contradicts (iv) of 1.3.1

1.3.1 seem completely correct except that in (i) it says P(A) = 1, while that should be P(S) = 1.

I cannot make much sense out of 1.3.5 - except by assuming there are additional restrictions on A, B & C, which are not shown in the excerpt.