Please explain each step carefully.
Please explain each step carefully. I. Let μ and v be sigmafinite measures on the measurable...
(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...
Show step by step please, I need A, B and C, THANKS! Let (V,<,>) be a finite-dimensional Euclidean space n and let T be linear operator in V. Are the following statements true? Show your answer. (show by and =) A. T is orthogonal if and only if t preserves angles, that is, if e is the angle between a and B, then 0 is the angle between T(a) and T(B). B. T is orthogonal if and only if T...
Let and be two finite measures on . Prove that if and only if the condition implies , for each . Thank you for your explanations. We were unable to transcribe this imageWe were unable to transcribe this image(N, P (N)) μ<<φ 6({n})=0 ({n}) = 0 neN
Please explain every step! And please write clearly. Thank you! Let V be a the shape described by the following criteria in cylindrical coordinates: 0 < z <r, Find the average square of the distance from the origin of V.
(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...
Please solve the exercise 3.20 . Thank you for your help ! ⠀ Review. Let M be a o-algebra on a set X and u be a measure on M. Furthermore, let PL(X, M) be the set of all nonnegative M-measurable functions. For f E PL(X, M), the lower unsigned Lebesgue integral is defined by f du sup dμ. O<<f geSL+(X,M) Here, SL+(X, M) stands the set of all step functions with nonnegative co- efficients. Especially, if f e Sl+(X,...
(6) Let (, A,i) be a measure space. Let fn : 0 -» R* be a sequence of measurable functions. Let g, h : O -> R* be a pair of measurable functions such that both are integrable on a set A E A and g(x) < fn(x)<h(x), for all E A and ne N. Prove that / lim sup fn du fn dulim sup fn du lim inf fn du lim inf n o0 A n-oo A noo n00...
Let M be a ơ-algebra of subsets of X and P the set of finite measures on M. Prove that (a) d(μ, v) = supAEM |μ( A)-v(A)| defines a metric on P (b) (P, d) is complete. Let M be a ơ-algebra of subsets of X and P the set of finite measures on M. Prove that (a) d(μ, v) = supAEM |μ( A)-v(A)| defines a metric on P (b) (P, d) is complete.
(6) Let (2,A, /i) be a measure space. Let fn: N -» R* be a sequence of measurable functions. Let g, h : 2 -> R* be a integrable pair of measurable functions such that both are on a set AE A and g(x) < fn(x) < h(x), for all x E A and n e N. Prove that / / fn du lim sup fn d lim sup lim inf fn d< lim inf fn du п00 n oo...
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)