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MeasureTheory2005Jan - Analysis Exam 29 January 2005...

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Unformatted text preview: Analysis Exam 29 January 2005 Measure Theory Part 1. Let f(1:) be the standard Cantor function. Define 9(1‘) 2 f(:c)+:c. Show that g is continuous, increasing, and 1-1 from [0,1] onto [0,2]. Use 9 to show that the image of a Lebesgue measurable set under a continuous map may not be measurable. 2. Consider the real line with Lebesgue measure. A sequence of measurable real valued functions fn converges in measure to the mea- surable function f. In addition [nt S g for all n where g is an integrable function. Show that ' hm/un—fl:0 3. Suppose that 1 < p < q < r < 00 and that f 6 UV) LT. Estimate the L" norm of f in terms of a product involving the LP and LT norms. Something like [|f[[q S ]|f[[$‘[[f]];‘a where 0 < (1 <1. 4. Let f be measurable on the interval [0,1] (Lebesgue measure on the real line). If the function g(3:,y) = $(f2(113) — f4(y)) is integrable on the unit square in R2 show that f is integrable on [0,1]. ...
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