5) Let E be measurable with m(E) < ∞. Suppose each fr : ER is measurable and fnf pointwise on E to some f: E R. Given any e > 0 and 8 > 0, prove (with- out using Egoroff's theorem) that there exists ACE and NE N where m(E\A) < € and \fn(x) f(x)\<8 for every Є A and n > N. Hint: Consider the sets AN = {x = E:\fn(x) - f(x)| < 6 for all n > N}. Prove these are ascending, UN-1AN E, and use continuity of measure. Now simply set A = AN for large enough N! =

5) Let E be measurable with m(E) < ∞. Suppose each fr : ER is measurable and fnf pointwise on E to some f: E R. Given any e > 0 and 8 > 0, prove (with- out using Egoroff's theorem) that there exists ACE and NE N where m(E\A) < € and \fn(x) f(x)\<8 for every Є A and n > N. Hint: Consider the sets AN = {x = E:\fn(x) - f(x)| < 6 for all n > N}. Prove these are ascending, UN-1AN E, and use continuity of measure. Now simply set A = AN for large enough N! =

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.3: The Natural Exponential Function

Problem 51E

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