Integration: Exercises BR Ch 11

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

Last updates: 2 April 2011

Integration: Exercises BR Ch 11

  1. If f0 and E fdμ =0, prove that f(x)=0 almost everywhere on E. Hint: Let En be the subset of E on which f(x)>1/n. Write A=En. Then μ(A)=0 if and only if μ(En)=0 for every n.
  2. If A fdμ =0 for every measurable subset A of a measurable set E, then f(x) =0 almost everywhere on E.
  3. If {fn} is a sequence of measurable functions, prove that the set of points x at which {fn(x)} converges is measurable.
  4. If fL1(μ) on E and g is bounded and measurable on E, then fgL1 (μ) on E.
  5. Put
    g(x) = { 0, if 0x 12, 1, if 12 x 1,
    f2k (x) = g(x), for 0x1, and f2k+1 (x) = g(1-x), for 0x1.
    Show that
    liminfn fn(x), for 0x1,
    01 fn(x) dx = 12.
  6. Let
    fn(x) = { 1n, if |x|n, 0, if |x|>n.
    Then fn(x) 0 uniformly on 1, but
    - fndx =2, for n>0 .
    Thus uniform convergence does not imply dominated convergence in the sense of Theorem 11.32. However on sets of finite measure, uniformly convergent sequences of bounded functions do satisfy Theorem 11.32.
  7. Find a necesary and sufficient condition that f (α) on [a,b]. Hint: Consider Example 11.6(b) and Theorem 11.33.
  8. If f on on [a,b] and if F(x) = ax f(t) dt , prove that F(x) =f(x) almost everywhere on [a,b].
  9. Prove that the function F given by (96) is continuous on [a,b].
  10. If μ(X)< and fL2(μ), prove that fL1(μ). If μ(X)= this is false. For instance, if m is Lebesgue measure on 1 and
    f(x) = 1 1+|x| , for x ,
    then fL2(m), but fL1(m).
  11. If f,g L1(μ), define the distance between f and g by
    X |f-g| dμ .
    Prove that L1(μ) is a complete metric space.
  12. Suppose
    (a)   |f(x,y) | 1, if 0x1, 0y1,
    (b)   for fixed x, f(x,y) is a continuous function of y,
    (c)   for fixed y, f(x,y) is a continuous function of x.
    g(x) = 01 f(x,y) dy ,    for 0x1.
    Is g continuous?
  13. Consider the functions
    fn(x) =sinnx ,    for n>0 and -πxπ,
    as points of L2(m). Prove that the set of these points is closed and bounded, but not compact.
  14. Prove that a complex function f is measurable if and only if f-1 (V) is measurable for every open set V in the plane.
  15. Let be the ring of all elementary subsets of (0,1]. If 0<ab1, define
    ϕ([a,b]) = ϕ([a,b)) = ϕ((a,b]) = ϕ((a,b)) =b-a ,
    but define
    ϕ((0,b)) = ϕ((0,b]) = 1+b ,
    if 0<b1. Show that this gives an additive set function ϕ on , which is not regular and which cannot be extended to a countably additive set function on a σ-ring.
  16. Suppose {nk} is an increasing sequence of positive integers and E is the set of all x(-π,π) at which {sinnkx} converges. Prove that m(E)=0. Hint: For every AE,
    A sinnkx dx 0 ,
    2 A (sinnkx) 2 dx = A (1- cos2nkx) dx m(A)      as n.
  17. Suppose E (-π,π), m(E)>0, δ>0. Use the Bessel inequality to prove that there are at most finitely many integers n such that sinnxδ for all xE.
  18. Suppose fL2(μ) and gL2(μ). Prove that
    | f g dμ | 2 = |f| 2 dμ |g| 2 dμ
    if and only if there is a constant c such that g(x) =cf(x) almost everywhere. (Compare Theorem 11.35.)

Notes and References

These exercises are taken from [RuB, Chapt. 11] for a course in "Measure Theory" at the Masters level at University of Melbourne.


[RuB] W. Rudin, Principles of Mathematical Analysis, Third edition, McGraw-Hill, 1976. MR??????.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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