Integration: Exercises R Ch 3

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

Last updates: 18 April 2011

Integration: Exercises R Ch 3

  1. Prove that the supremum of any collection of convex functions on (a,b) is convex on (a,b) (if it is finite) and that pointwise limits of sequences of convex functions are convex. What can you say about upper and lower limits of sequences of convex functions?
  2. If φ is convex on (a,b) and if ψ is convex and nondecreasing on the range of φ, prove that ψφ is convex on (a,b). For φ>0, show that the convexity of logφ implies the convexity of φ, but not vice versa.
  3. Assume that φ is a continuous real function on (a,b) such that
    φ ( x+y2 ) 12 φ(x) + 12 φ(y)
    for all x and y (a,b). Prove that φ is convex. (The conclusion does not follow if continuity is omitted from the hypothesis.)
  4. Suppose f is a complex measurable function on X, and
    φ(p) = X |f| p dμ = fpp (0<p< ).
    Let E= {p | φ(p)<}. Assume f >0.
    (a)   If r<p<s, rE, and sE, prove that pE.
    (b)   Prove that logφ is convex in the interior of E and that φ is continuous on E.
    (c)   By (a), E is connected. Is E necessarily open? Closed? Can E consist of a single point? Can E be any connected subset of (0,)?
    (d)   If r<p<s, prove that fp max( fr , fs ). Show that this implies the inclusion Lr(μ) Ls(μ) Lp(μ) .
    (e)   Assume that fp < for some r< and prove that
    fp f as p .
  5. Assume, in addition to the hypotheses of Exercise 4, that
    (a)   Prove that fr fs if 0<r<s.
    (b)   Under what conditions does it happen that 0<r<s and fr = fs <?
    (c)   Prove that Lr(μ) Ls(μ) if 0<r&t;s. Under what conditions do these two spaces contain the same functions?
    (d)   Assume that fr < for some r>0, and prove that
    limp0 fp = exp { X log|f| dμ }
    if exp{-} is defined to be 0.
  6. Let m be Lebesgue measure on [0,1], and define fp with respect to m. Find all functions Φ on [0,) such that the relations
    Φ( limp0 fp ) = 01 (Φf) dm
    holds for every bounded, measurable, positive, f. Show first that
    cΦ(x) + (1-c) Φ(1) = Φ(xc) (x>0, 0c1) .
    Compare with Exercise 5(d).
  7. For some measures, the relation r<s implies Lr(μ) Ls(μ) ; for others the inclusion is reversed; and there are some for which Lr(μ) does not contain Ls(μ) if rs. Give examples of these situations, and find conditions on μ under which these situations will occur.
  8. If g is a positive function on (0,1) such that g(x) as x0, then there is a convex function h on (0,1) such that hg and h(x) as x0. True or false? Is the problem changed if (0,1) is replaced by (0,) and x0 is replaced by x.
  9. Suppose f is Lebesgue measurable on (0,1) and not essentially bounded. By Exercise 4(e), fp as p. Can fp tend to arbitrarily slowly? More precisely, is it true that to every positive function Φ on (0,) such that Φ(p) as pinfin; one can find an f such that fp as p, but fp Φ(p) for all sufficiently large p?
  10. Suppose fn is in Lp(μ), for n=1,2,3, , and fn-fp 0 and fng a.e., as n. What relation exists between f and g?
  11. Suppose μ(Ω)=1, and suppose f and g are positive measurable functions on Ω such that fg1. Prove that
    Ω fdμ Ω gdμ 1 .
  12. Suppose μ(Ω)=1 and h:Ω [0,] is measurable. If
    A= Ω hdμ,
    prove that
    1+A2 Ω 1+h2 dμ 1+A .
    If μ is Lebesgue measure on [0,1] and if h is continuous, h=f, the above inequalities have a simple geometric interpretation. From this, conjecture (for general Ω) under what conditions on h equality can hold in either of the above inequalities, and prove your conjecture.
  13. Under what conditions on f and g does equality hold in the conclusions of Theorems 3.8 and 3.9? You may have to treat the cases p=1 and p= separately.
  14. Suppose 1<p<, fLp =Lp( (0,)), relative to Lebesgue measure, and
    F(x) = 1x 0x f(t) dt (0<x<) .
    (a)   Prove Hardy's inequality
    Fp pp-1 fp .
    which shows that the mapping fF carries Lp into Lp.
    (b)   Prove that equality holds only if f=0 a.e.
    (c)   Prove that the constant p/(p-1) cannot be replaced by a smaller one.
    (d)   If f>0 and f L1, prove that F L1.
    Suggestions: (a) Assume first that f0 and fCc ((0,)). Integration by parts gives
    0 Fp(x) dx = -p 0 Fp-1(x) x F(x) dx .
    Note that xF =f-F, and apply Hölder's inequality to Fp-1 f. Then derive the general case. (c) Take f(x) =x-1/p on [1,A], f(x)=0 elsewhere, for large A. See also Excercise 14, Chap. 8.
  15. Suppose {an} is a sequence of positive numbers. Prove that
    N=1 ( 1N n=1N an ) p ( pp-1 ) p n=1 anp
    if 1<p<. Hint: If an an+1, the result can be made to follow from Exercise 14. This special case implies the general one.
  16. Prove Egoroff's theorem: If μ(X) <, if {fn} is a sequence of complex measurable functions which converges pointwise at every point of X, and if ε>0, there is a measurable set EX, with μ (X-E)<ε such that {fn} converges uniformly on E.
    (The conclusion is that by redefining the fn on a set of arbitrarily small measure we can convert a pointwise convergent sequence to a uniformly convergent one; note the similarity with Lusin's theorem.)
    Hint: Put
    S(n,k)= i,j>n { x| |fi(x) - fj(x)| < 1k ,
    show that μ(S(n,k)) μ(X) as n, for each k, and hence that there is a suitably increasing sequence { nk} such that E=S(nk, k) has the desired property.
    Show that the theorem does not extend to σ-finite spaces.
    Show that the theorem does extend, with essentially the same proof, to the situation in which the sequence {fn} is replaced by a family {ft}, where t ranges over the positive reals; the assumptions are now that, for all xX,
    (i)   limt ft(x) =f(x) and
    (ii)   tft(x) is continuous.
  17. (a)   If 0<p<, put γp= max(1, 2p-1) , and show that
    |α-β| p γp ( |α|p + |β|p )
    for arbitrary complex numbers α and β.
    (b)   Suppose μ is a positive measure on X, 0<p<, fLp(μ), fn Lp(μ), fn(x) f(x) a.e., and fnp fp as n. Show that then lim f-fn p =0 , by completing the two proofs that are sketched below.
    (i)   By Egoroff's theorem, X=AB in such a way that A |f|p <ε , μ(B)<, and fnf uniformly on B. Fatou's lemma, applied to B |f|p , leads to
    limsup A |fn|p dμ ε .
    (ii)   Put hn= γp ( |f|p + |fn|p ) - |f-fn|p , and use Fatou's lemma as in the proof of Theorem 1.34.
    (c)   Show that the conclusion of (b) is false if the hypothesis fnp fp is omitted, even if μ(X)<.
  18. Let μ be a positive measure on X. A sequence {fn} of complex measurable functions on X is said to converge in measure to the measurable function f if to every ε>0 there corresponds an N such that
    μ( {x | |fn(x) -f(x)| >ε} ) <ε
    for all n>N. (this notion is of importance in probability theory.) Assume μ(X) < and prove the following statements:
    (a)   If fn(x) f(x) a.e., then fnf in measure.
    (b)   If fn Lp(μ) and fn-f p 0, then fnf in measure; here 1p.
    (c)   If fnf in measure, then {fn} has a subsequence which converges to f a.e.
    Investigate the converses of (a) and (b). What happens to (a), (b), and (c) if μ(X) =, for instance, if μ is Lebesgue measure on 1?
  19. Define the essential range of a function fL (μ) to be the set Rf consisting of all complex numbers w such that
    μ( {x | | f(x)-w| <ε} ) >0
    for every ε>0. Prove that Rf is compact. What relation exists between the set Rf and the number f?
    Let Aj be the set of all averages
    1μ(E) E fdμ
    where E and μ(E)>0. What relations exist between Af and Rf? Is Af always closed? Are there measures μ such that Af is convex for every f L(μ)? Are there measures μ such that Af fails to be convex for some f L(μ)?
    How are these results affected if L(μ) is replaced by L1(μ), for instance?
  20. Suppose φ is a real function on 1 such that
    φ( 01 f(x) dx ) 01 φ(f) dx
    for every real bounded measurable f. Prove that φ is then convex.
  21. Call a metric space Y a completion of a metric space X if X is dense in Y and Y is complete. In Sec. 3.15 reference was made to "the" completion of a metric space. State and prove a uniqueness theorem which justifies this terminology.
  22. Suppose X is a metric space in which every Cauchy sequence has a convergent subsequence. Does it follow that X is complete? (See the proof of Theorem 3.11.)
  23. Suppose μ is a positive measure on X, μ(X)<, fL(μ), f >0, and
    αn = X |f|n dμ (n=1,2,3, ) .
    Prove that
    limn αn+1 αn = f .
  24. Suppose μ is a positive measure, fLp(μ), gLp(μ).
    (a)   If 0<p<1, prove that
    | |f|p - |g|p | dμ |f-g|p dμ ,
    that Δ(f,g) = |f-g|p dμ defines a metric on Lp(μ), and that the resulting metric space is complete.
    (b)   If 1p< and fp R, gp R, prove that
    | |f|p - |g|p | dμ 2pRp-1 f-gp .
    Hint: Prove first, for x0, y0, that
    | xp-yp | { |x-y|p, if  0<p<1, p|x-y| (xp-1 + yp-1), if  1p<.
    Note that (a) and (b) establish the continuity of the mapping f |f|p that carries Lp(μ) into L1(μ).
  25. Suppose μ is a positive measure on X and f:X (0,) satisfies Xfdμ =1. Prove, for every EX with 0<μ(E)< , that
    E (logf)dμ μ(E) log 1μ(E)
    and, when 0<p<1,
    E fp dμ μ(E) 1-p .
  26. If f is a positive measurable function on [0,1], which is larger,
    01 f(x) logf(x) dx or 01 f(s) ds 01 logf(t) dt ?

Notes and References

These exercises are taken from [Ru, Chapt. 3] 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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