Let (X, A, µ) and (Y, B, v) be measure spaces. Prove that (a) if E E A x B then y-section of E is measurable. (b) if f : X ×Y→R is a measurable function then y-section of f is measurable.

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A: The given in this question is a measurable space (X,F) and two F-measurable functions f and g from X…

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Q: The set of functions S = span{e™" | n e Z} separates [0,1]). True False

A: S=span{enx |n∈Z}

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Q: (b) Give an example of a bounded function f on [a,b] for which f2 e R[a,b), but f is not in R[a,b).

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Q: "Prove that the function f : R→ R with ƒ(x) = = is uniformly continuous."

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Q: Theorem (6) If f is bounded and R – I on [a, b], f(x) > 0, Vx € [a, b] then ſ f 2 0

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Q: Ja (16) Let f e Lla, b]. Show that if g is a bounded measurable function then fa E Cla bl

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Q: Let f(x) be a function continuous on [1, 3] and differentiable on (1, 3), such that f(1) = 1, f(3) =…

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Q: lim f is measurable.

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Q: Let the function ƒ : [0, 1] →R be defined by { if x = ! (n = 1, 2, 3, · · · ) f(x) = n 1– x2…

A: Given :  f : [0, 1] →ℝ  defined by ;              fx = ex2,if  x=1n    (n=1, 2, 3, ...…

Q: If f(x, y, z) = = identify the domain and range of f(x, y, z). Describe, in words, the set of points…

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