Question

Please show all details. State any lemmas, propositions and theorems used. Include all details. Make sure it is legible. Thanks!

Let f be a non-negative measurable function. Show that there is a sequence (Pn) of non-negative simple functions, each of which vanishes outside of a set of finite measure, such that f = lim An.

Answer 1

. let us suppose that f 7,0 Fear neNa we define on as to = select if 44 3456x) <** ,::1,2,--sha? in if f(xit, Then an (1) s f(x) Let € be domen off t h e De fire fins = { kee: done 5 1600 szins En = { 166 : $622215 Then each of sets is measurable since f is mensweable. We note that bn = ( 2 ) Keni + hlen for every hen since ch. fn of a meastaable set is meastable a n is meastaable for each new mosareng ban can takes only a finite no of values and $270 THEN. And in is a non-decreasing sequence of nen-negative simple fus. na

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