|(16) Let (, A, ) be a measure space and let f finite a.e -> R* be integrable. Prove that f is |(16) Let (, A, ) be a measure space and let f finite a.e -> R* be integrable. Prove that f is
(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...
complete measure space (i.e. ВЕА, "(В) — 0, АсВ — АЄ (5) Let (Q, A, м) be a A, u(A) = 0). Let f,g : Q+ R* be a pair functions. Assume that f is measurable g almost everywhere and that f (a) Prove that g is measurable (b) Let A E A and assume that f is integrable on A. Prove that g is integrable on A and g du complete measure space (i.e. ВЕА, "(В) — 0, АсВ...
(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...
(1) Let (2, A, i) be a measure space {AnE A E A} is a (a) Fix E E A. Prove that Ap 0-algebra of E, contained in A. (b) Let /i be the restriction of /u to Ap. Prove that ip is a measure on Ap. (c) Suppose that f : O -» R* is measurable (with respect to A). Let g the restriction of f to E. Prove that g : E -> R* is measurable (with respect...
(1) Let (, A, /i) be a measure space = {AnE: A E A} is a o-algebra of E, contained in (a) Fix E E A. Prove that AE A. (b) Let be the restriction of u to AE. Prove that uE is a measure on Ag (c) Suppose that f -> R* is measurable (with respect to A). Let g = f\e be the restriction of f to E. Prove that g E ->R* is measurable (with respect to...
(1) Let (, A, i) be a measure space. {AnE: Ae A} is a o-algebra of E, contained in (a) Fix E E A. Prove that Ap = A. (b) Let uE be the restriction of u to AĘ. Prove that iE is a measure on Ag. (c) Suppose that f : Q -» R* is measurable (with respect to A). Let g = the restriction of f to E. Prove that g : E ->R* is measurable (with respect...
Let (, A, ) be a measure space. Let fn : 2 -» R* be a sequence of measurable functions. Let g,h : 2 -» R* be a pair of measurable functions such that both are integrable on that a set A E A and g(r) fn(x)h(x), for all E A and nE N. Prove / fn du lim sup A lim inf fn dulim inf lim sup fn du A fn du no0 no0 A noo n+o0 (You may...
(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...
(5) Let (N, A, д) be a complete measure spaсе (i.e. В € А, д(В) — 0, А с В — АЄ A, uA 0. Let f,g : 2 -> R* be a pair functions. Assume that f is measurable and that f g almost everywhere (a) Prove that g is measurable. (b) Let A E A and assume that f is integrable on A. Prove that g is integrable on A and f du g ар. A A (5)...