Integration: Exercises

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 5 March 2011

Integration: Exercises

  1. Does there exist an infinite σ-algebra which has only countably many members?
  2. Prove an analogue of [Ru, Theorem 1.8] for n functions.
  3. Prove that if f is a real function on a measurable space X such that {x| f(x)r} is measurable for every rational r, then f is measurable.
  4. Let {an} and {bn} be sequences in [-,] and prove that
    (a) limsup n (-an) = - liminf n an .
    (b) limsup n (an+bn) limsup n an + limsup n bn , provided none of the sums is of the form -. Show by an example that strict inequality can hold.
    (c) If an bn for all n then liminf n an liminf n bn .
  5. (a)   Suppose f:X [-,] and g:X [-,] are measurable. Prove that the sets
    {x| f(x)< g(x) } and {x| f(x)= g(x) }
    are measurable.
    (b)   Prove that the set of points at which a sequence of measurable real-valued functions converges (to a finite limit) is measurable.
  6. Let X be an uncountable set, let be the collection of all sets EX such that either E or Ec is at most countable, and define μ(E)=0 in the first case, μ(E)=1 in the second. Prove that is a σ-algebra on X and that μ is a measure on . Describe the corresponding measurable functions and their integrals.
  7. Suppose fn :X[0,] is measurable for n=1,2,3,,    f1 f2 f3 0 ,    fn(x) f(x) as n for every xX,    and    f1 L1(μ). Prove that then
    limn X fn dμ = X f dμ
    and show that this conclusion does not follow if the condition "f1 L1(μ)" is omitted.
  8. Put fn=χE if n is odd, and fn=1-χE if n is even. What is the relevance of this example to Fatou's lemma?
  9. Suppose μ is a positive measure on X, f:X[0,] is measurable, and X f dμ =c where 0<c< and α is a constant. Prove that
    limn X n log(1+ (f/n) α) dμ = { , if 0<α<1, c, if α=1, 0, if 1<α<.
    Hint: If α1, the integrands are dominated by αf. If α<1, Fatou's lemma can be applied.
  10. Suppose μ(X)<, fn is a sequence of bounded complex measurable functions on X, and fnf uniformly on X. Prove that
    limn X fn dμ = X f dμ .
  11. Show that
    A= n=1 k=n Ek
    in [Ru, Theorem 1.41], and hence prove the theorem without any reference to integration.
  12. Suppose fL1(μ). Prove that to each ϵ>0 there exists a δ>0 such that X |f| dμ <ϵ whenever μ(E)<δ.
  13. Show that [Ru, Proposition 1.24(c)] is also true when c=.

Notes and References

These exercises are taken from [Ru] for a course in "Measure Theory" at the Masters level at University of Melbourne.


[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR??????.

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