Counting measure problem (integrals)

Definition: For any E in X, where X is any set, define M(E) = infinity if E is an infinite set, and let M(E) be then umber of points in E if E is finite. M is called the counting measure on X. Let f(x) : R -> [0,infinity) f(j) = { a_j , if j in Z, a if j in RZ} ( Z here is counting numbers, R is set of real numbers) Let M be the counting measure. Find integral over R of f dM. My thoughts on this one: I thought about starting here by finding a simple function then take the integral of it, then after that find Sup to get integral of f. ( I believe that we can show that f is measurable function, also, it is a positive function since its range is [0,infinity) ).

Integrals of measurable functions

Let X be an uncountable set, let m be the collection of all sets E in X such that either E or E^c is at most countable, and define M(E) = 0 in the first case, and M(E) = 1 in the second case. ( m here is sigma algebra in X). The Questions is : Describe the integrals of the corresponding measurable functions.

Q on Lebesgue integrals.

In a previous problem I posted here: Let f(x) be a positive continuous function on [0,1/2], f(x) =< 1/2. Let A = { (x,y) : 0 =< x = 1/2, 0=

Reimann and Lebesgue integrals.

(a) If f is a nonnegative continuous function on [0,1], then show that integral from 0 to 1 f(x) dx = integral over [0,1] f dx ( that is show that the reimann integral and lebesgue integrals are equal). (b) Prove part (a) for any continuous function.

Fixed point of a compressing function on metric space

Fixed point of a compressing function on metric space See attached file for full problem description with symbols.