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) ...continues
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.
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.
--- 1. Give an example of a set E such that both E and its complement are dense in R^1. Then show that such a set E can not be closed. Note: we are using the "Methods of Real Analysis by Richard R Goldberg" ---
2. Prove that if a metric space M is totally bounded, then there is a countable dense subset of M. Note: we are using the "Methods of Real Analysis by Richard R Goldberg
(See attached file for full problem description with proper equations) --- 3. Let Show that T is a contraction on (0. ,but that T has no fixed point on this interval. Does this conflict Theorem 6.4? Explain. Note: We are using the book Methods of Real Analysis by Richard R. Goldberg. This theorem 6.4 is ...continues
(See attached file for full problem description with proper equations) --- 6. Let be a totally bounded metric space, and is uniformly continuous and onto. Show is totally bounded. Note: we are using the "Methods of Real Analysis by Richard R Goldberg" ---
Please can you explain me with more detail about Lebesgue measure of Q. Why m(Q)=0 and m(In)=2/n. (See attached file for full problem description)
