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Real Analysis Final Exam 2008-1-8 Name ID Show detailed argument to each problem. 1. 10 points Let E Rn be a measurable set and q x 0 be a nonnegative measurable function on E. Show that for any constant 0 we have the inequality Z qdx q 1 dx. E solution By H lder inequality q 1 Z dx the proof is done. 2. 10 points Assume E and u is a measurable function on E which is everywhere positive. For 0 p set 1/p p p u u x dx. E E Assume that 1. up x 1 uniformly on E as p 0. R d p R p d 2. dp u x...

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