Hi, I'm wondering if someone can help me with the following questions relating to the following definitions:

Given sample space S with event space A probability measure is defined as:

1) Non-Negative Function, P: A-->[0, infinity)

2) Normed, P(S)=1

3) Sigma-Additive, A = A1+...+An+...

4) Finitely Additive

5) Sequentially Continuous, An ---> A => P(An) ---> P(A)

Now, I need to prove the following:

1) Prove that if we suppose 1), 2), 4) and 5) are true, then 1), 2) and 3) must be true.

2) Given that An--->A and Bn--->B prove that An*Bn--->AB and Complement(An)--->Complement(A) and $\displaystyle A_n \bigcup B_n --> A \bigcup B$

3) We say (6) P is sequentially continuous at 0 iff $\displaystyle A_n \downarrow \varnothing => P(A_n) \downarrow 0$

Prove that 1),2),4) and 5) <==> 1), 2), 4) and 6)

Thanks a lot