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# hw9 - | g x | ≤ M a.e Comment If g is essentially bounded...

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Math 318 HW #9 Due 5:00 PM Thursday, April 14 Reading: Wilcox & Myers § 28–30. Problems: 1. Let A be a bounded, measurable set and let ( f n ) be a sequence of measurable functions on A converging to f . Suppose ϕ L ( A ) and that | f n ( x ) | ≤ ϕ ( x ) for all x A and all n = 1 , 2 , . . . . Show that lim n →∞ A f n g dm = A fg dm if g is measurable and essentially bounded on A , meaning that there exists M > 0 such that
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Unformatted text preview: | g ( x ) | ≤ M a.e. ( Comment: If g is essentially bounded, then the quantity ess sup x ∈ A | g ( x ) | := inf Z ⊂ A m ( Z )=0 ± sup x ∈ A \ Z | g ( x ) | ² , called the essential supremum of A , is ﬁnite.) 2. Exercise 31.16. 3. Exercise 31.33. You may ﬁnd Problem 1(b) from HW 8 useful. 1...
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