hw9 - | g ( x ) | M a.e. ( Comment: If g is essentially...

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Math 318 HW #9 Due 5:00 PM Thursday, April 14 Reading: § 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 →∞ Z A f n g dm = Z 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 ) | , called the essential supremum of A , is nite.) 2. Exercise 31.16. 3. Exercise 31.33. You may nd Problem 1(b) from HW 8 useful. 1...
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