Integration: More Exercises

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

Last updates: 18 March 2011

Integration: More Exercises

  1. If μ is a complex measure on a σ-algebra , and if E, define λ(E) = sup|μ (Ei) |, the supremum being taken over all finite partitions {Ei} of E. Does it follow that λ=|μ|?
  2. Prove that the example given at the end of [Ru, Sec. 6.10] has the stated properties.
  3. Prove that the vector space M(X) of all complex regular Borel measures on a locally compact Hausdorff space X is a Banch space if μ = |μ|(X). Hint: Compare [Ru, Ch. 5, Exercise 8]. [That the difference of any two members of M(X) is in M(X) was used in the first paragraph of the proof of [Ru, Theorem 6.19]; supply a proof of this fact.]
  4. Suppose 1p and q is the exponent conjugate to p. Suppose μ is a positive σ-finite measure and g is a measurable function such that fg L1(μ) for every f Lp(μ). Prove that then g Lq(μ).
  5. Suppose X consists of two points a and b; define μ({a})=1, μ({b}) = μ(X)=, and μ()=0. Is it true, for this μ that L(μ) is the dual space of L1(μ)?
  6. Suppose 1<p< and q is the exponent conjugate to p. Prove that Lq(μ) is the dual space of Lp(μ) even if μ is not σ-finite.
  7. Suppose μ is a complex Borel measure on [0,2π) (or on the unit circle T), and define the Fourier coefficients of μ by
    μ^ (n)= e-iπt dμ(t) (n=0, ±1,±2,) .
    Assume that μ^ (n)0 as n+ and prove that then μ^ (n)0 as n-. Hint: The assumption also holds with fdμ in place of dμ; if f is any trigonometric polynomial, hence if f is continuous, hence if f is any bounded Borel function, hence if dμ is replaced by d|μ|.
  8. In the terminology of Exercise 7, find all μ such that μ^ is periodic, with period k. [This means that μ^(n+k) = μ^(n) for all integers n; of course, k is also assumed to be an integer.]
  9. Suppose that {gn} is a sequence of positive continuous functions on I =[0,1], that μ is a positive Borel measure on I, and that
    (a) lim n gn(x) =0, a.e.[m],
    (b) I gn dm =1 ,    for all n,
    (c) lim n I fgn dm = I f dμ ,    for every fC(I).
    Does it follow that μm?
  10. Let (X,,μ) be a positive measure space. Call a set Φ L1(μ) uniformly integrable if to each ε<0 corresponds a δ>0 such that
    | E f dμ | <ε ,
    whenever fΦ and μ(E)<δ.
    (a)   Prove that every finite subset of L1(μ) is uniformly integrable.
    (b)   Prove the following convergence theorem of Vitali: If
    (i)   μ(X)<,
    (ii)   {fn} is uniformly integrable,
    (iii)   fn f(x) a.e. as n,
    (iv)   |f(x)| < a.e.,
    then f L1(μ) and
    lim n X |f-fn| dμ =0 .
    Suggestion: Use Egoroff's theorem.
    (c)   Show that (b) fails if μ is Lebesgue measure on (-,), even if {fn 1} is assumed to be bounded. Hypothesis (i) can therefore not be omitted in (b).
    (d)   Show that hypothesis (iv) is redundant in (b) for some μ ( for instance, for Lebesgue measure on a bounded interval), but that there are finite measures for which the omission of (iv) would make (b) false.
    (e)   Show that Vitali's theorem implies Lebesgue's dominated convergence theorem, for finite measure spaces. Construct an example in which Vitali's theorem applies although the hypotheses of Lebesgue's theorem do not hold.
    (f)   Construct a sequence {fn}, say on [0,1], so that fn(x) 0, but {fn} is not uniformly integrable (with respect to Lebesgue measure).
    (g)   However, the following converse of Vitali's theorem is true: If μ(X)<, fn L1(μ) and lim n E fn dμ exists for every E, then {fn} is uniformly integrable.
    Prove this by completing the following outline.
    Define ρ(A,B) = |χA-χB | dμ . Then (,ρ) is a complete metric space (modulo sets of measure 0), and E E fn dμ is continuous for each n. If ε>0, there exist E0,δ,N (Exercise 13, Chapt. 5) so that
    | X (f-fn) dμ | <ε ,    if   ρ(E,E0) <δ,     n>N. (*)
    If μ<δ, (*) holds with B= E0-A and C=E0A in place of E. Thus (*) holds with A in place of E and 2ε in place of ε. Now apply (a) to {f1, f2, fn}: There exists δ>0 such that
    | X fn dμ | <3ε ,    if   μ(A) <δ,     n=1,2,3,.
  11. Suppose that μ is a positive measure on X, μ(X), fn L1(μ) for n=1,2,3, , fn f(x) a.e., and there exists p >1 and C< such that X |fn|p dμ <C for all n. Prove that
    lim n X |f-fn| dμ =0 .
    Hint: {fn} is uniformly integrable.
  12. Let be the collection of all sets E in the unit interval [0,1] such that either E or its complement is at most countable. Let μ be the counting measure on this σ-algebra . If g(x)=x for 0x1, show that g is not -measurable, although the mapping
    f xf(x) =fg dμ
    makes sense for every fL1 (μ) and defines a bounded linear functional on L1 (μ). Thus (L)* L1 in this situation.
  13. Let L= L(m), where m is Lebesgue measure on I= [0,1]. Show that there is a bounded linear functional Λ0 on L that is 0 on C(I), and that therefore there is no gL1(m) that satisfies Λf= I fg dm for every fL. Thus (L)* L1.

Notes and References

These exercises are taken from [Ru, Chapt. 6] 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. MR0924157.

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