Bressan problems

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 18 March 2014

Bressan problems

  1. Let {Ai|iI} be an open covering of a compact metric space K. Prove that there exists ρ>0 such that if xK then there exists jI such that B(x,ρ)Aj.
  2. Let x1,x2, be a sequence in a metric space E. Prove the following.
    1. The sequence x1,x2, converges to x if and only if from every subsequence (xnj)j>0 one can extract a further subsequence converging to x.
    2. If d(xm,xn)δ where δ>0 for all mn then no convergent subsequence can exist.
    3. Let E be complete and assume that, for every ε>0, from any sequence once can extract a further subsequence (xnj)j>0 such that limsupj,kd (xnj,xnk) <ε. Then the sequence admits a convergent subsequence.
  3. Consider the function f(x)= { 1x(lnx)2, if0<x<12, 0, otherwise. Let fεJε*f be the corresponding mollifications, and let F(x)sup0<ε<1 fε(x). Prove that fL1() but FL1(). As a consequence, although fεf pointwise, one cannot use the Lebesgue dominated convergence theorem to prove that fε-fL1()0.
  4. Let fn: for n1, be a sequence of absolutely continuous functions such that
    1. at the point x=0, the sequence fn(0) is bounded,
    2. there exists a function gL1() such that the derivatives fn satisfy |fn(x)|g(x) for every n1 and a.e. x.
    Prove that there exists a subsequence (fnj)j1 which converges uniformly on the entire real line.
  5. Consider a sequence of functions fnL1() with fnL1C for every n1. Define f(x) { limnfn(x), if the limit exists, 0, otherwise. Prove that f is Lebesgue measurable and fL1C.
  6. Let f: be an absolutely continuous function. Prove that f maps sets of Lebesgue measure zero into sets of Lebesgue measure zero.
    1. If (fn)n>0 is a sequence of functions in L1([0,1]) such that fnL10 prove that there exists a subsequence that converges pointwise for a.e. x[0,1].
    2. Construct a sequence of measurable functions fn:[0,1][0,1] such that fnL10 but, for each x[0,1] the sequence fn(x) has no limit.
  7. For every (nonempty) open set Ωn and for 1p, prove that the space Lp(Ω) is infinite dimensional. Construct a sequence of functions (fj)j>0 such that fjLp = 1,and fi-fjLp 1for alli,j>0 withij.
  8. Consider the set 2 with the partial ordering xy if and only if x1y1 and x2y2. Let f: be a continuous, non decreasing function. Show that the set S=Graph(f)= {(t,f(t))|t} is a maximal totally ordered subset of 2. Is every maximal totally ordered subset obtained in this way?
  9. Give a proof of the generalized Hölder inequality: If 1p1,,pm and 1p1++1pm=1 and fkLpk(Ω) for k=1,2,,m then Ω|f1f2fm| dxk=1m fkLpk(Ω).

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