Problem 3. Let f be a nonnegative measurable function on R. Show that imd. 72 IR
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove that for any A E A f du lim fn du A 4 (You must show that the integrals exist.) (3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove...
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, : O -> R* be a sequence of measurable functions such that g(x) < fn(x) < fn+1(x), for all E A and n E N. Prove that lim fn d lim fn du noo A (11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function...
(5) Let (. A, /u) be a measure space. Let f,g : O > R* be a pair functions. Assume that f is measurable and that f = g almost everywhere. (a) Prove that q is measurable on A. Prove that g is integrable (b) Let A E A and assume that f is integrable on A and A (5) Let (. A, /u) be a measure space. Let f,g : O > R* be a pair functions. Assume that...
(4) Let (Ω,A) be a measurable space, and let f : Ω → R. Prove that the following statements are equivalent: ·f is measurable. ·f-1(1) E A for any open interval I c R. lei f (A) E A for any open set ACR ·f-1 (A) E A for any Borel set A c R. (4) Let (Ω,A) be a measurable space, and let f : Ω → R. Prove that the following statements are equivalent: ·f is measurable. ·f-1(1)...
(4) Let (Q,A) be a measurable space, and let f : Ω-> R. Prove that the following statements are equivalent: f is measurable . f-(I) E A for any open interval I CR .f-(A) E A for any open set ACR. . f-(A) E A for any Borel set ACR. (4) Let (Q,A) be a measurable space, and let f : Ω-> R. Prove that the following statements are equivalent: f is measurable . f-(I) E A for any open...
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and compute the value of f du (4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and...
(a) Let Ω = [4, 101 and let A = 16, 6], [8, 10]} 2. (i) Find F(A) (ii) Let X : 2->R be defined by X = 2-1[4,5]-3 . 1 (6,8) Is X, F(A)-measurable? Justify your answer. (b) Let (2, F) be a measurable space, and let X :2-R. Suppose that X+ is F-measurable. Does this imply that X is F-measurable? Either prove it or give a counterexample. (a) Let Ω = [4, 101 and let A = 16,...
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I = [0, 1] and compute the value of f du (4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I...
Problem 4. Let w be a positive continuous function and let n be a nonnegative integer. Equip P.(R) with the inner product (p,q) = $' p(x)q(x)"(x) dx. You do not need to check that this is an inner product. (a) Prove that P.(R) has an orthonormal basis po..., Pr such that deg pk = k for each k. (b) Show that (Pk, pk) = 0 for each k, where the polynomials pį are from the preceding part. Here pé denotes...
Please explain each step carefully. I. Let μ and v be sigmafinite measures on the measurable space (X,M). Suppose v << μ and let f be a nonnegative M-measurable function on X. Prove [du