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