4. Prove that the external measure is not "additive", in the sense that [0, 1]le <...
Problems 5) Let (X, M, u) be a measure space, and f e Lt. Assume that S fdu = 1. Prove that 00, 0<a<1, lim n ln (1 +(${2))a) du(x) = { 1, a = 1, 10. a 1. Hint: Use Fatou's lemma for a < 1 and LDCT for a > 1 (dominate by af). 1+00)
3) Prove that there exists f : R → R non-negative and continuous such that f € L'OR : dm) ( i.e. SR \f|dm <00) and lim sup f(x) = ∞. 2-0
Bartle The Elements of Integration and Lebesgue Measure: 4.R. If fe M*(X, X) and is due < +00, then the set N = {xe X: f(x) > 0} is o-finite (that is, there exists a sequence (Fa) in X such that N CU Fn and u(F.) < too).
8 arbitrary set. K is Cousider E} n=1 nieU and Let (X, K) be a measure space where X is an sigma-algebra of subsets of X and is a measure sequenc o clemenis of K We delin lim supn(Fn) liminfn(En)- U then prove: (a) lim in(E)) lim inf(u(E,) (b) T J (c) If sum E,)x, then (lim sup(E)) = 0 x X) <oc lor somc nE N, then lim supn (Fn)> lim sup(u(F,n )) 8 arbitrary set. K is Cousider...
Prove that if ? is integrable on [?, ?] and ?(?) ≥ 0 for all ? in [?, ?], then [ f(x)dx > 0 7. Prove that if f is integrable on [a, b] and f(x) > 0 for all x in [a, b], then sof(x)dx > 0.
real analysis. questions Prove that if lima In = 0 and > M for some M >0 and in 10 > 0, then lima (ny) - Asume 30 = 2,2-20+ In+1 = In + Prove that this sequence has a limit and find the limit. Prove that lim = L with L < if and only if every subsequence limo n L. Suppose that the sequence {an) is increasing and the sequence {yn) is decreasing. Moreover, lim a n -...
1 0 4. Consider the matrices A = 0 +- Alw alcaldo and B o -1010 = 01. Answer the following o 0 2 questions. (5) Find all the vectors x and y which satisfy the following simultaneous equations. y = lim {A^ + B” k} n >00 \y\=1. Here, y is the length of the vector y.
Please prove this, thanks! 2. Let {xn n21 be a sequence in R such that all n > 0. If ( lim supra) . (lim supー) = 1 Tn (here we already assume both factors are finite), prove that converges.
let X be s random nareprion if x <0 > 0 (a) Let M= {X > 1). Find Fx( M)
3. (a) Given n e N, prove that sup{.22 : 0<x<1} = 1 and inf{.22n: 0<x<1} = 0. (b) Find the supremum of the set S = {Sn: ,ne N}. Give a proof.