Real analysis qualifying exam
August 2012
Each problem is worth ten points. Work each problem on a separate piece of paper.
1. Let (X, M, µ)be a measure space. Prove that the normed vector space L1(X, µ)is complete.
You may use any results except the convergence of function series.
2. Fix two measure spaces (X, M, µ)and (Y, N, ν )with µ(X), ν(Y)>0. Let f:X→C,
g:Y→Cbe measurable. Suppose f(x) = g(y) (µ⊗ν)-a.e. Show that there is a constant a∈C
such that f(x) = a µ-a.e. and g(y) = a ν -a.e.
3. Let f:R3→Rbe a Borel measurable function. Suppose for every ball B,fis Lebesgue
integrable on Band RBf(x)dx = 0. What can you deduce about f? Justify your answer carefully.
4. Let Xbe a locally compact Hausdorff space. Denote by C0(X)the space of complex-valued
continuous functions on Xwhich vanish at infinity, and by Cc(X)the subset of compactly sup-
ported functions. Use an appropriate version of the Stone-Weierstrass theorem to prove that Cc(X)
is dense in C0(X).
5. Give an example of each of the following. Justify your answers.
(a) A nowhere dense subset of Rof positive Lebesgue measure.
(b) A closed, convex subset of a Banach space with multiple points of minimal norm.
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