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  3. dct dominated convergence theorem proposition 710 the set of...

Question: dct dominated convergence theorem proposition 710 the set of...

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DCT = Dominated Convergence Theorem

Proposition 7.10: The set of simple functions is dense in Lp(S)   (p in [1,infty) and let S be a set)

Problem 3 (Improving Corollary 7.12 from the lecture notes). Let (S, d) be metric space and let μ be a measure on (S,B(S)) such that μ(S) oo. Let p E [1,00) (a) Let B B(S). Show that for every 0 there is an open set G C S such that SSI (b) Let G be an open set. Define fn S0, oo) by fn (s)- min[nd(s, G), 1) for n 2 1. Then by Proposition 5.4 from Carothers book we know d(,Ge) is continuous and thus that fn is continuous. Use DCT to show |lf 1gllp -> 0 (c) Let f E LP(S) and ε > 0. Show that there exists a bounded continuous function -glp < e? g E Cb(S) such that |If Hint: First useProposition 7.10 to approximate f bv a simple function, and then use (a) and (b) in a suitable way

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