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real analysis - elements of integration

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real analysis - elements of integration

Hi everybody,
Can you helpto solve these questions
1)Show that the following functions are measure, if they are not measure explain why

a)set x is N=naturals, X is sigma algebra of all subsets of N
a(n) is a sequence of non negative real numbers, and
f(empty set)=0 , f(E)=(SUM a(n)) where sum is over n is element of E, where E is nonempty

b)set x be uncountable set and X be the family of all subsets of x then let f(E)=0 if E is countable, f(E)=+infinity if E is uncountable

c)set x is N=natural numbers, X is family of all subsets of N ,
f(E)=0 if E is finite, f(E)=+infinity if E is infinite

d)f(E)=sup ( sum from j=1 to n |g(A(j))| ) where supremum is taken over all finite disjoint collections {A(j)} in X with E=union j=1 to n (A(j)) ,where g is a charge on X


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