Real Analysis : Jacobians of Trigonometric Functions.
Compute the Jacobian where
Lebesgue Measurable Sets, Compact Sets and Open Sets
If A is lebesgue measurable sets in R^n, bounded, then there is a compact set K_epsilon and an open set for every epsilon > 0 V_epsilon such that K_epsilon is subset of A and A is a subset of V_epsilon and for m(A-K) < epsilon m(V-K) < epsilon
Let A be a set in R^n, we denote by A + x_o a parallel shift of A by x_o to A + x_o, A + x_o = { x : x = y + x_o, y in A}. Now, if A is a lebesgue measurable then show that 1). x_o + A is also lebesgue measurable 2). m(A) = m(x_o + A) Can someone check my answer and tell me if it is correct or not? My work: s ...continues
Measurable Sets and Spaces and Properties of Integrals of Simple Functions
1).If A is a subset of B, A,B in m ( measurable sets) then show that integral (region A) s dM =< integral ( region B) s dM Where s here is a simple non-negative measurable function. ( Please don't confuse this with bounded measurable functions, I need the proof for SIMPLE functions). 2). If E are measurable, X_E is the c ...continues
(See attached file for full problem description with proper symbols) --- Let and for (a) Use integration by parts to show that in for . Deduce that for (b) Compute for and verify that ---
True or False problem. m_* (A) = Sup sum_i | M_i| ( U M_i is subset of A) Where m_* is the inner measure M_i doesn't equal M_j for i doesn't equal j ( i.e, they are disjoint) Prove it or show a counterexample and explain it to show how the equality doesn't hold.
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 number 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.
