Let ([0, 1], L, m) be a Lebesgue measure space, and let A be a nonempty measurablesubset of [0, 1]. Let {E} [0, 1] be a countable disjoint collection of Lebesguemeasurable sets.a. Show thatm (AnŨE₂) = Σm(ANE).k=1k=1b. Let ƒ : [0, 1] → (0, 1] be a measurable function. Show that for every € > 0,Nethere is a natural number N₁ and a set C such that m(C₂) < € and №²+1 < f(x) ≤ / /for all x € Ce.

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Answered: Let ([0, 1], L, m) be a Lebesgue…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let ([0, 1], L, m) be a Lebesgue measure space, and let A be a nonempty measurable subset of [0, 1]. Let {E} [0, 1] be a countable disjoint collection of Lebesgue measurable sets. a. Show that m(AnŨEx) = Σm(An Ek). k=1 k=1