 # (1) Suppose that f1 : [0,1] R and f2 : (1, 2] R are measurable func...

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(1) Suppose that f1 : [0,1] R and f2 : (1, 2] R are measurable functions. Define X € = X € Show that F is a measurable function on [0,2]. (2) Let E C R have the property that m. (EnK) = 0 for all compact subsets K CR. Prove that m. (E) = 0. (3) Given an integrable function f on R, prove that limn-+0 JEn f = SRF where En = {x € R : f(x) (4) Assume that A CRd has the property that AnK is measurable for all compact subsets K C Rd. Verify that A is measurable. (5) Let S be the line segment {(t,t) : t € [0,1]}. Show that S has Lebesgue measure zero in the plane. (6) Let f be a nonnegative measurable function on R. Show that if SRf = 0, then f = 0 almost everywhere. (7) Suppose that E is a bounded measurable set. Show that for any € > 0 there is a finite collection of disjoint intervals I1,I2, IN such that N m < €. n=1 (8) Given an integrable function on R², show that Colp x=0 (((2,y) dy) dx = So (I 0(2,y) da) dy y=0 x (9) Let f and g be bounded functions on R. Verify that sup[f(x + g(x)] < sup f (x) + sup g(x). xER xER xER (10) Show that for any integrable function § on the interval [0,1 1] we have 1 lim d(x) sin(2mnx) dx = 0. n->oo

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