Functions : L-Spaces ( Lebesgue Spaces )
Consider the following function: f(x) = 1/x for x in [1, infinity) = 1 for x in (-1,1) = -1/x for x in (-infinity, -1] Please explain why f(x) is in L^2(R)L^1(R)
Suppose X is a measurable space, E belongs to the sigma algebra ( I believe to the sigma algebra in X) , let us consider XE = Y. Show that all sets B which can be expressed as AE, where A belongs to the sigma algebra in X, form a sigma-algebra in Y. Please justify every step and claim you make in the solution.
Real Analysis Jacobians(I) Explanation of the condition - not independent of the Jacobians of functions.
Real Analysis Topology and Sigma-Algebra
1). Prove that any sigma-algebra, which contains a finite number of members is also a topology. ( The Q in another words : to show that there exist a sequence of disjoint members of a sigma algebra which contains infinite no. of members). 2). Does there exist an infinite sigma-algebra which has only countably many members? ...continues
Real Analysis : Measurable sets and functions
1).If f: X--> C ( C is complex plane) is measurable, then prove that f^-1({0}) ( f inverse of 0 or any other point) is a measurable set in X. 2). If E is measurable set in X and if X_E ( x) = { 1 if x is in E, 0 if x is not in E} then X_E is a measurable function. Now I want you to prove the other direction, that is, I w ...continues
Suppose u(x) : X--> R v(x) : X --> R Both u(x) and v(x) are measurable Let f(x) : x --> R^2 f(x) = (u(x), v(x) ) Then f (x) is measurable Now prove a generalization of the above. That is, prove: if u_1(x) : X--> R u_2(x): X--> R . . . . u_n(x) : X--> R u_1,. ...continues
See attached Show that u,v,w are not independent. Also find the relation between u,v and w .
Real Analysis Jacobians (II) Explanation of the condition - not independent of the Jacobians of functions.
If f is one-to-one, f, f^-1 are continuous, then f is called a homeomorphism. Now I want you to prove the following: Let f : X -> Y, ( X and Y are topological spaces)be homeomorphism, prove that it establishes one-to-one correspondence between Borel sets in X and Y.
Real Analysis, sup, inf, measurable functions.
------------------------------------------------------------------------------------------- 1). If g_n = Sup f_n, then prove that ( g_n)^-1 ( ( alpha, infinity] ) = union ( n = 1 to infinity) (f_n)^-1((alpha,infinity]). ------------------------------------------------------------------------------------------- 2). Pr ...continues
