If E M [a, b], we define the (Lebesgue) measure of E to be the number m(E)
Question:
(a) Show that m(θ) = 0 and 0 (b) Show that m([c, d]) = m([c, d)) = m((c, d]) = m((c, d)) = d - c.
(c) Show that m(Eʹ) = (b - a) - m(E).
(d) Show that m(E F) + m(E © F) = m(E) + m(F).
(e) If E © F = θ show that m(E F) = m(E) + m(F). (This is the additivity property of the measure function.)
(f) If (Ek) is an increasing sequence in M [a, b], show that m [k=1 Ek) = limk(Ek). [Use the Monotone Convergence Theorem.]
(g) If (Ck) is a sequence in M [a, b] that is pairwise disjoint (in the sense that Cj © Ck = θ, whenever j k), show that
(This is the countable additivity property of the measure function.)
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Related Book For
Introduction to Real Analysis
ISBN: 978-0471433316
4th edition
Authors: Robert G. Bartle, Donald R. Sherbert
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