## Urgent: Messurable set theory

Hello there I have two question which I have been stuck with these last couple of weeks.

(a)

Let (E, \epsilon) messurable space and m: \epsilon -> \mathbb{R}_{x} be finite additive.

I need to show that m is increasing and that m(\empty) = 0.

I need to conclude that m is a messure, if and only if m is either m is upwards continous, or m is countable subadditive.

(b)

Let F be a non-empty set and let (B_n)_{n \geq 1} be elements in 2^F.

Here I need to show

lim_n sup B_n = lim_n inf B_n = \bigcup_{n=1} ^{\infty} B_n

if B_n \subseteq B_{n+1} for all n \geq 1, and that

lim_n sup B_n = lim_n inf B_n = \bigcap_{n=1} ^{\infty} B_n

if B_n \supseteq B_{n+1} for all n \geq 1.

I then assume that F = \mathbb{R} and that B_n = [0,y_n], where (y_n)_{n \geq 1} is a limited sequence of positive real numbers.

Finally I need to show

[0, lim_n sup y_n [ \subseteq lim_{n} sup B_n \subseteq [0, lim_n sup y_n]

I need some emergency help because my Dad is very ill (heart trouble), and I need to get finished with this here within the next 5 hours, so I can go look after him.

I know this is much to ask, but is there anybody here who could help show (a) and (b) ??

Thank You and God Blees.

Best Regards,

Linda21