Integration: Exercises Chapter 2

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

Last updates: 20 March 2011

Integration: Exercises Chapter 2

  1. Let {fn} be a sequence of real nonnegative functions on 1 and consider the following four statements:
    (a)   If f1 and f2 are upper semicontinuous, then f1+f2 is upper semicontinuous,
    (b)   If f1 and f2 are lower semicontinuous, then f1+f2 is lower semicontinuous,
    (c)   If each fn is upper semicontinuous, then 1 fn is upper semicontinuous,
    (d)   If each fn is lower semicontinuous, then 1 fn is lower semicontinuous,
    Show that three of these are true and one is false. What hapens if the word "nonnegative" is omitted? Is the truth of the statements affected if 1 is replaced by a general topological space?
  2. Let f be an arbitrary function on 1, and define
    ϕ(x,δ) = sup{ |f(s) -f(t) | s,t (x-δ,x+δ) },
    ϕ(x) = inf{ ϕ(x,δ) | δ>0 },
    Prove that ϕ is upper semicontinuous, that f is continuous at a point x if and only if ϕ(x)=0, and hence that the set of points of continuity of an arbitrary complex function is a Gδ.
    Formulate and prove an analogous statement for general topological spaces in place of 1.
  3. Let X be a metric space, with metric ρ. For any nonempty EX, define
    ρE(x) = inf{ρ((x,y) | yE}.
    Show that ρE is a uniformly continuous function on X. If A and B are disjoint nonempty closed subsets of X, examine the relevance of the function
    f(x) = ρA(x) ρA(x) +ρB(x)
    to Urysohn's lemma.
  4. Examine the proof of the Riesz theorem and prove the following two statements:
    (a)   If E1V1 and E2V2, where V1 and V2 are disjoint open sets, then μ(E1 E2) = μ(E1+ E2) , even if E1 and E2 are not in .
    (b)   If EF, then E= N K1 K2 , where {Ki} is a disjoint countable collection of compact sets and μ(N) =0.
  5. Let m denote Lebesgue measure on 1. Let E be Cantor's familiar "middle thirds" set. Show that m (E)=0 even though E and 1 have the same cardinality.
  6. Let m denote Lebesgue measure on 1. Construct a totally disconnected compact set K1 such that m(K)>0. (K is to have no connected subset consisting of more than one point.)
    If v is lower semicontinuous and v χK, show that actually v0. Hence χK cannot be approximated from below by lower semicontinuous functions, in the sense of the Vitali-Carathéodory theorem.
  7. Let m denote Lebesgue measure on 1. If 0<ϵ<1, construct an open set E[0,1] which is dense in [0,1], such that m(E)=ϵ. (To say that A is dense in B means that the closure of A contains B.)
  8. Let m denote Lebesgue measure on 1. Construct a Borel set E1 such that
    0<m( EI)< m(I)
    for every nonempty segment I. Is it possible to have m(E)< for such a set?
  9. Construct a sequence of continuous functions fn on [0,1] such that 0fn 1, such that
    limn 01 fn(x) dx=0 ,
    but such that the sequence {fn(x) } converges for no x[0,1].
  10. If {fn} is a sequence of continuous functions fn on [0,1] such that 0fn 1 and such that fn(x)0 as n for every x [0,1], then
    limn 01 fn(x) dx=0 .
    Try to prove this without using any measure theory or any theorems about Lebesgue integration. (This is to impress you with the power of the Lebesgue integral. A nice proof was given by W.F. Oberlein in Communications on Pure and Applied Mathematics, vol. X, pp. 357-360, 1957.)
  11. Let μ be a regular Borel measure on a compact Hausdorff space X; assume that μ(X)=1. Prove that there is a compact set KX (the carrier or support of μ) such that μ(K)=1 but μ(H)<1 for every proper comapct subset H of K. Hint: Let K be the intersection of all compact Kα with μ( Kα)=1; show that every open set V which contains K also contains some Kα. Regularity of μ is needed; compare exercise 18. Show that Kc is the largest open set in X whose measure is 0.
  12. Show that every compact subset of 1 is the support of a Borel measure.
  13. Is it true that every compact subset of 1 is the support of a continuous function? If not, can you describe the class of all compact sets in 1 which are supports of continuous functions? Is your description valid in other topological spaces?
  14. Let f be a real-valued Lebesgue measurable function on k. Prove that there exist Borel functions g and h such that g(x) =h(x) a.e. [m], and g(x)f(x) h(x) for every x k.
  15. It is easy to guess the limits of
    0n (1- xn) n ex/2 dx       and       0n (1+ xn) n e-2x dx
    as n. Prove that your guesses are correct.
  16. Why is m(Y)=0 in the proof of Theorem 2.20(e)?
  17. Define the distance between points (x1,y1) and (x2,y2) in the plane to be
    | y1-y2 |    if x1 =x2,        1+ | y1-y2 |    if x1 x2.
    Show that this is indeed a metric, and that the resulting metric space X is locally compact.
    If f Cc(X), let x1, xn be those values of x for which f(x,y) 0 for at least one y (there are only finitely many such x!), and define
    Λf= j=1n - f(xj,y) dy .
    Let μ be the measure associated with this Λ by Theorem 2.14. If E is the x-axis, show that μ(E)= although μ(K)=0 for every compact KE.
  18. Let X be a well-ordered uncountable set which has a last element ω1, such that every predecessor of ω1 has at most countably many predecessors. ("Construction": Take any well-ordered set which has elements with uncountably many predecessors, and let ω1 be the first of these; ω1 is called the first countable ordinal.) For αX, let Pα (resp. Sα) be the set of all predecessors (resp. successors) of α, and call a subset of X open if it is a Pα or an Sβ or a Pα Sβ or a union of such sets. Prove that X is a compact Hausdorff space. (Hint: No well ordered set contains an infinite decreasing sequence.)
    (a)   Prove that the complement of the point ω1 is an open set which is not σ-compact.
    (b)   Prove that to every fC(X) there corresponds an αω1 such that f is constant on Sα.
    (c)   Prove that the intersection of every countable collection {Kn} of uncountable compact subsets of X is uncountable. (Hint: Consdier limits of increasing countable sequences in X which intersect each Kn in infinitely many points.)
    Let be the collection of all EX such that either E{ω1} or Ec {ω1} contains an uncountable compact set; in the first case, define λ(E)=1; in the second case, define λ(E)=0. Prove that is a σ-algebra which contains all Borel sets in X, that λ is a measure on which is not regular (every neighborhood of ω1 has measure 1), and that
    f(ω1) = X f dλ .
    for every fC(X). Describe the regular μ which Theorem 2.14 associates with this linear functional.
  19. Go through the proof of Theorem 2.14, assuming X to be compact (or even compact metric) rather than just locally compact, and see what simplifications you can find.
  20. Find continuous functions fn: [0,1][0 ,) such that fn (x)0 for all x[0,1] as n, 01 fn(x) dx0 , but supnfn is not in L1. (This shows that the conclusion of the dominated convergence theorem may hold even when part of its hypothesis is violated.)
  21. If X is compact and f: (-,) is upper semicontinuous, prove that f attains its maximum at some point of X.
  22. Suppose that X is a metric space, with metric d, and that f:X [0,] is lower semicontinuous f(p)< for at least one pX. For n= 1,2,3, and xX, define
    gn(x) = inf{ f(p)+nd( x,p) | pX }
    and prove that
    (i)   |gn(x) - gn(y)| nd(x,y),
    (ii)   0 g1 g2 f ,
    (iii)   gn(x) fn(x) as n, for all xX.
    Thus f is the pointwise limit of an increasing sequence of continuous functions. (Note that the converse is almost trivial.)
  23. Suppose that V is open in k and μ is a finite positive Borel measure on k. Is the function that sends x to μ(V+x) necessarily continuous? lower semicontinuous? upper semicontinuous?
  24. A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in 1. Assume fL1 (1), and prove that there is a sequence {gn} of step functions so that
    limn - |f(x) - gn(x) | dx =0 .
  25. (i)   Find the smallest constant c such that
    log(1+et) <c+t (0<t<) .
    (ii)   Does
    limn 1n 01 log(1+e nf(x)) dx .
    exist for every real fL1? If it exists, what is it?

Notes and References

These exercises are taken from [Ru, Chapt. 2] 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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