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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