S. Take p E (1,0o) and let p' be the conjugate exponent of p. Let K
6. Suppose that m is o -finite. Let p and r be real numbers such that
a. Show that the map it: f H fg is continuous from LP to L'.
Hint. Show that otherwise there exists a sequence (fn)n>1 of positive functions of LP such that, for every n > 1, IIfnIIp < 1 and IIfn91ir > n.
Then prove, on the one hand, that the function h = En' n-2 fn is in Lp/r and therefore that f = h'/r is in L", and on the other hand that f g ¢ Lr.
b. Deduce that g E LQ, where q is given by 1/r = 1/p+ 1/q.
Hint. Let (An)nEN be an increasing sequence of Cements of F with finite measure and such that UnEN An = X. Put
9n = (inf(I9I,n)) IA,,.
Show that
1/r `1/p
(J9dm) <I4II(J 9dm f
7. An ordered set (E, is called d a conditionally complete lattice if every nonempty subset of E that has an upper bound has a supremum (least upper bound) in E, and every nonempty subset that has a lower bound has an infimum in E.
We consider the space E = L", for 1 < p:5 oo, with the natural order defined by
f < g f_-s f (x) < g(x) m-almost everywhere.
a. Suppose p = 1. Let ii be a nonempty family in LR bounded above, and let V be the set of its upper bounds.
i. Show that the expression a = inf {f f dm : f E `l' } is finite.
ii. Show that there exists a decreasing sequence (fn) in * such that lim / fn dm = a.
n-a+oo
Let f be the almost-everywhere limit of (fn). Show that f E V and that f f dm = a.
ill. Deduce that f is the supremum of 0 in LR, and so that Ls is a conditionally complete lattice.
Hint. If g E V, show that f inf (f, g) dm = a and deduce that f <g-
b. Suppose that 1 < p < oo. Show that LR is a conditionally complete lattice.
Hint. If srd is a nonempty family in L°a bounded above, the set If Iflp-1 : f E d} is contained in LR.
152 4. LP Spaces
c. i. Show that if m is or-finite Ll is a conditionally complete lattice.
Hint. Start by dealing with the case where m has finite mass (then LOO C L').
ii. Show that this result may be false if m is not a-finite.
Hint. Take two uncountable disjoint sets A and B. Let X be their union, let 9 be the set of subsets of X that are countable or have countable complement, let m be the count measure on
9, and set 0 _ (1())XEA
d. Let E be the quotient of the space of s-measurable real functions by the relation of equality m-almost everywhere. Give E the natural order defined earlier. Show that, if m is a-finite, E is a conditionally complete lattice.
Is the space of 9-measurable real functions with the natural order a conditionally complete lattice?
8. Prove that the set of integrable piecewise constant functions is dense in Ll if and only if m has finite mass.
Hint. Take f = 1. If m has infinite mass, any integrable piecewise con- stant function s lies at a distance 11s - f 11 > 1 from f.
9. Prove that L' n L°° is dense in Ll if and only if m has finite mass.
10. Consider the following property:
(P) There exists an (infinite) sequence of .f-measurable, pairwise dis- joint subsets of X of positive measure.
a. Show that, if (P) is satisfied, Ll is not separable.
Hint. You can use as inspiration the e°O case in Exercise 7 on page 11.
b. Suppose (P) is not satisfied. Define an atom as any .''-measurable subset A of positive measure that does not contain any subset B E 9 with m(B) > 0 and m(A \ B) > 0.
i. Show that every measurable subset of X with nonzero measure contains at least one atom.
Hint. Consider the relation < defined on the set d of elements of .$ of nonzero rn-measure by
A<B r-* m(B\A)=O.
Apply Zorn's Lemma (see Exercise 11 on page 133) to the or- der relation induced by < on the quotient set d ., where is equality almost everywhere:
A^-B r--> m(B\A)=m(A\B)=0 A<BandB<A.
You might show, in particular, that every totally ordered subset of 0jti has a greatest element.
H. Show that there exists a finite sequence (Xn)n<, of atoms such that m(X \ U,,<no 0 and
forn0m.
W. Show that every .%-measurable function coincides rn-almost ev- erywhere with a linear combination of functions 1X,,, for n < no.
c. Show the equivalence of the following properties:
i. (P) is not satisfied.
ii. LO0 has finite dimension.
iil. LOO is separable.
iv. Every F-measurable function belongs to 2O°.
11. Let L be a vector subspace of APR (m) fl 2R (m) satisfying these hy- potheses:
- There exists an increasing sequence (cpn) in L that converges to 1 in-almost everywhere.
- The o-algebra o(L) generated by L equals Sr.
- f2ELforallf EL.
a. Give the space LI (m) fl Lr(m) the norm Q + 1!°o and denote by L the closure of L in that space. Show that f E L implies If I E L.
Deduce that If E L for all f E L.
Hint. Use the example on page 29 and argue as in the proof of Theorem 2.3 on page 33.
b. Show that L is dense in L' (m).
Hint. Apply Proposition 2.4 on page 63.
c. Deduce that L is dense in LP (m) for 1 < p < oo.
Hint. If f E La (m), you might show that, for every n E N, the function sup (inf (f, n<p. ), -npn) can be approximated in LPR (in) by a sequence in L.
12. Let in and p be or-finite measures on measurable spaces (X, .9) and (Y, q), and suppose p E (1, oo). We denote by LP(m) ® LP(p) the vector subspace of LP(m x p) spanned by the functions (x, y) H f (x)g(y), with f E LP(m) and g E LP(p). Show that LP(m) ® LP(p) is dense in LP(m x p). This generalizes the result of Exercise 7 on page 110.
Hint. Apply the result of Exercise 11 above to the measure in x p and the space L = (.SeR(m) fl 2R (m)) ® (2j(p) n YR (p))
13. Assume in is o-finite.
a. Suppose the o-algebra .9T is separable, that is, generated by a count- able family of subsets of X.
I. Show that there exists a countable family 9 of elements of F satisfying these conditions:
- o(t) = 9, where o(.) is the a-algebra generated by .4.
- Af1BE5dforallA,BER.
- m(A) < +oo for all A E R.
- There exists an increasing sequence of elements of . whose union equals X.
154 4. L' Spaces
H. Show that the family {lA}AE. is fundamental in LP, for 1 <
p<00.
Hint. Apply Exercise 11 above.
Hi. Deduce that, if 1 < p < oo, the space LP is separable.
iv. Show that, if X is a separable metric space, the Borel a-algebra R(X) is separable. Derive hence another proof for Corollary 1.7.
b. We say that a a-algebra..9' is almost separable if there exists a sep- arable a-algebra 5' contained in .1F such that, for all A E 9, there exists B E .' with
m(A \ B) = m(B \ A) = 0.
i. Show that, if Jr is almost separable, the space LP is separable for every p E [1,00).
Hint. Use part a.
ii. Show that if there exists p E 11, oo) such that LP is separable, . ' is almost separable.
Hint. Consider the a-algebra generated by a sequence of ele- ments of 2P whose corresponding classes are dense in LP.
iii. Show that Jr is almost separable if and only if there exists a sequence (An)nEN of measurable subsets of X of finite measure such that the sequence is fundamental in V.
iv. Let 5f be the set of elements of 9 of finite measure, modulo the relation of equality m-almost everywhere. If A, B E 5f, we write d(A,B) = m(AAB), where A A B = (AUB) \ (AflB).
Show that d makes 5f into a complete metric space, separable if and only if the a-algebra 9 is almost separable.
Hint. (5f, d) can be identified with the subset of L' consisting of (classes of) characteristic functions of elements in Jr, with the metric defined by the norm [I iii.
14. Assume p E [1,00).
a. Let Y be the set of finite families (An)n<no in 9 such that - m(AnflA*n)=0 if n# m, and
- 0 <m(An) <oo for every n < no.
If W = (An)n<_n0 is an element of Y, we define an operator Td on LP by
Turf = l 1 f f dm 1 A.
n<no m(An) /
Show that T& is a continuous linear operator on LP, of norm at most 1.
b. If at and .4 are elements of Y, write 0 C M if every element of .4 is contained, apart from a set of measure zero, in an element of d,
and if every element of W is, apart from a set of measure zero, the union of the elements of 9 contained in it.
Let n0 = (A,,),,<, be an element of 9 and let f be a linear com- bination of functions 1A,,, for n < no. Show that, for every 9 E 9 such that W C 9, we have T5e f = f. Deduce that, for every e > 0 and every f E LP, there exists 0 E 9 such that
(..E9and WE:R)=* IITTf- fIlp<e.
Hint. Use the fact that the set of integrable piecewise constant func- tions is dense in L" (see the remark on page 146).
c. Assume that m has finite mass and that there exists a sequence (&0n) of 9 increasing with respect to C and such that Wo = {X}. Assume also that WEN on generates 9 (you can check that there is such a sequence if the a-algebra F is separable: see Exercise 13). Denote by .Son the set of piecewise constant functions that are constant on each element of din. Show that Un Yn is dense in LP for I <_ p < oo. (You could use Exercise 11, for example). Deduce that, for every f E La, the sequence (Td f) converges to f in II.
Example. Choose for X the interval [0,1], for m the Lebesgue mea- sure on X, and for 9 the Borel v-algebra of X. Find a sequence
(s fn) satisfying the conditions stated above.
15. We say that a sequence (fn)nEN of F-measurable functions converges in measure to a F-measurable function f if, for every e > 0,
m({x E X : I fn(x) - f (x)l > e}) -+ 0.
a. Assume p E 11, oo).
L Bienaymc-Chebyshev inequality. Take f E L. Show that, for every b > 0,
m({x E X : If (x)I > S}) < b-PlI f 11p,
ii. Let (fn) be a sequence of elements of L" that converges in L" to f E L". Show that the sequence (fn) converges to f in measure.
b. Let (fn) be a sequence of measurable functions that converges in measure to a measurable function f.
i. Show that there exists a subsequence such that, for every
kEN,
m({x E X : I fnr(x) - f(x)I > 2-k}) < 2-k.
ii. For each k E N, let Zk be the subset of X defined by Zk = U {x E X :I fnj(x)- f(x)I > 2-j}.
j>k
Then set z =' IkENZk. Prove that m(Z) = 0.
156 4. LP Spaces
iii. Deduce that the sequence (f, ) converges to f m-almost every- where.
iv. Show also that, for every e > 0, there exists a measurable subset A of X of measure at most a and such that the sequence (f n,, ) converges uniformly to f on X \ A.
Hint. Choose A = Zk, with k large enough.
c. Suppose m(X) < +oo. Let (fn) be a sequence of measurable func- tions that converges m-almost everywhere to a measurable function f. Show that the sequence (fn) converges in measure to f.
Hint. Take e > 0. For each integer N E N, put
AN = {xeX:Ifn(x)-f(x)I <e for all n > N}.
Show that there exists an integer N E N for which m(X \ AN) < e, and therefore that m(X \ An) < e for every n > N.
Deduce that, for every integer n > N,
m({x E E : I fn(x) - f (x)I > e}) < C.
16. Suppose p E [1, oo]. Let (fn)nEN be a sequence in LP such that the series EnEN Ilfn - fn+iIlp converges. Show that the sequence (fn) converges almost everywhere and in LP.
Hint. Suppose first that m(X) is finite and prove that in this case J F, Ifn - fn+1I dà < +oo.
nEN
If m is arbitrary and p < oo, check that the set {x E X : fn(x) 0 for some n E N} is or-finite.
17. Equiintegrability. Assume p E [1, oo). A subset 3t° of LP is called equi- integrable of order p if for every e > 0 there exists 6 > 0 such that, for every measurable subset A of X of rn-measure at most b,
for all f E. °.
a. i. Show that every subset .fir of LP for which
lim J I f (P dm = 0 uniformly with respect to f E Je (*) Ill>nl
is equiintegrable of order p. Deduce that every finite subset of LP is equiintegrable of order p.
Show that, conversely, every bounded subset Ji° of LP that is equiintegrable of order p satisfies (*).
ii. Take Jr C I.P. Suppose there exists an element g E LP, nonneg- ative rn-almost everywhere, such that, for every f E Jr, we have If 1:5 g m-almost everywhere. Show that Jr is equiintegrable of order p.
iii. Let U101104 be a sequence in LP that converges in LP to f. Show that the family (fn)fEN is equiintegrable of order p.
Hint. You might check that, if A is a measurable subset of X, then
1/p r 11/p
(jIfI"dm) <IIf-fnIIP+(JAIfIPdm) .
b. We now assume that m has finite mass.
L Let (fn)nEN be a sequence in LP and let f E LP. Show that the sequence (fn)fEN converges to f in LP if and only if these conditions are satisfied:
The sequence (fn)nEN converges in measure to f (see Exercise 15 above for definition).
- The family { fn}nEN is equiintegrable of order p.
ii. Let (fn)nEN be a sequence of elements of LP that converges in measure to a function f. Assume that there exists g E LP such that IfnI < IgI for every n E N. Show that f E LP and that the sequence (fn) converges to f in LP.
iii. Let (fn)nEN be a bounded sequence in LP that converges almost everywhere to a function f. Check that f E LP. Then show that, for every real q E [l, p), we have Ilfn - f II Q = 0.
Hint. Note that if A is a measurable subset of X and if g E LP, then fA IgI°dm < IIgIIpm(A)1-9/P.
18. Uniformly convex spaces. A Banach space E is called uniformly convex if it has this property:
If (xn) and (yn) are sequences in the closed unit ball B(E) of E satisfying Ilxn + ynll -+ 2, then Ilxn - vnII -+ 0.
a. Show that every Hilbert space is uniformly convex.
b. Show that, for n > 2, the space Rn with the norm II II1 or the norm
11 II.., is not uniformly convex.
c. Let E be a uniformly convex space. Show that every nonempty con- vex closed subset of E contains a unique point of minimal norm.
d. Let E be a uniformly convex space.
I. Let f be a linear form on E of norm 1 and let (xn) be a sequence of elements of E of norm 1. Show that, if f (xn) -+ 1, the sequence (xn) converges.
Hint. You might show that (xn) is a Cauchy sequence, using the fact that f (xn + x,n) -+ 2 when n, m +oo.
158 4. L° Spaces
ii. Deduce that the absolute value of any continuous linear form on E attains its maximum in the closed unit ball of E.
e. Assume p E (1,00).
i. Show that
Ix2ilP< IxIP+IyIP forallx,yEC.
2 (*)
ii. Set D = {z E C : IzI < 1}. Show that the function W defined on D by
l+zP
(z) = 1 + IzIP
is continuous from D to [0,2P-1] and that So(z) = 2P-1 if and only if z = 1. Deduce that, for every r) > 0, there exists S(rl) > 0
such that, for every (x, y) E D2 with Ix - yI > rl, x+yr' < (1-6(i)) IxIP+IyIP
2 2
iii. Take e > 0 and let f and g be points in the closed unit ball of LP such that Ilf - gIIP > e. Set
E = {x E XJ : If(x) -9(x)I > e2-21pmax(If(x)I,I9(x)I)}.
A. Show that \Eif - glpdm < ep/2. Deduce that
f IfIP+I9Ip> sp
E 2 dm
2.2P (You might use (*) with x = f and y = -g.) B. Show that
+g P<1-b e "'+1
II 2 IIP \22/P/ 2Pwhere
6 is as in part e-ii above.
Hint. Use (*) in X \ E and the conclusion of e-ii in E, taking il = e/22/p.
C. Deduce that L' is uniformly convex (Clarkson's Theorem).
f. Let X be a metric space and give E = Cb(X) the uniform norm II '11.
Suppose that X contains a point a that is not isolated, and fix a sequence (xn) of pairwise distinct points in X that converges to a.
For f E E, put
nEN
i. Show that L is a continuous linear form on E of norm 2 and that JL(f) I < 2 for all f E B(E).
ii. Set C = (f E E : L(f) = 2}. Show that C is a nonempty dosed convex set in E, that 11f 11 > 1 for all f E C, and that
,nf fEC 11111= 1-
19. Suppose m is a Radon measure. If 1 < p < oo, we denote by L(m), or, more simply, by L ', the set of equivalence classes of functions f such that, for every compact K in X, the function 1K f lies in U. We denote by L' the set of elements of LP having compact support (the support of an element of LP was defined on page 145).
a. Show that, if 1 < p < q < oo, then L a C LP and L4 C L . b. Find a metric d on LP such that, for every sequence (fn)nEN in L oc
and every f E L P, the condition limner+. d(f, fn) = 0 is equivalent to the condition that lim,-,+,,111 K (fn - f) 11p = 0 for every compact K of X. Show that L ô is complete with this metric.
Hint. You might work as in Exercise 12 on page 57.
c. Show that the space LP, is dense in L oc with the metric d.
2 Duality
We consider again in this section a measure space (X, .$) and a measure m on 9. We assume here that m is o-finite. We will determine, for 1:5 p < oo, the topological dual (Lu)' of the space U.
So fix p E (1, +oo) and let p' be the conjugate exponent of p, so that 1 /p + 1 /p' = 1. Note first that every element g E LP' defines a linear form T. on LP, as follows:
Tgf= (fgdm forallfEL'. (*)
As an immediate consequence of the Holder inequality, the linear form T9 is continuous and its norm in (L')' is at most that of g in LP'. We will show that one obtains in this way all continuous linear forms on U.
Theorem 2.1 If 1 < p < oo, the linear map g H T. defined on LP' by (*) is a surjective isometry from 1? onto (1)')'.
If p = p' = 2, this is of course an immediate consequence of the Riesz Representation Theorem (Theorem 3.1 on page 111) in the Hilbert space L2. The basic scheme of the proof is to reduce the problem to this case.
This can easily be done if 1 < p < 2, but we will give a proof that is valid for every p E [1, oo), whose main idea goes back to J. von Neumann.
Proof. The proof of Theorem 2.1 will be carried out in several steps. The crucial point is the following lemma.
160 4. L' Spaces
Lemma 2.2 Suppose m has finite mass. Let T be a continuous linear form on L. If T is positive (that is, if T f > 0 for every f E LR such that f _> 0), there exists a measurable function g > 0 such that, for every f E LR,
f g E LR and T f= ff g dm.
Proof. (All functions are assumed real-valued without further notice.) Since the linear form T is positive, we can define on (X,.9) a measure A of finite mass by setting
A(A) = T(1 A) for all A E 9. (**)
That A is o-additive follows easily from the continuity and linearity of T (using the Dominated Convergence Theorem, which is allowed because m has finite mass). Then we set
v=A+m. (t)
Since T acts on classes of functions, we see that m(A) = 0 implies A(A) = 0;
thus, for A E F,
v(A) = 0 a m(A) = 0 A(A) = 0.
Hence the linear form f ti f f dA is well defined on L2(v) and we have, for every f E L2(v),
JfdA l <
(Jf2dA)1/2
(A(X)) 1/2 <_ IIf IIL2(r) (A(X))1/2.
By the Riesz Representation Theorem (Theorem 3.1 on page 111) applied to the Hilbert space L2(v), there exists an element h in L2(v) such that
JfdA =Jfhdv for allf EL 2 (V). ($) In particular,
0<A({h<0})= fh<o) hdv<0,
which implies that h > 0 v-almost everywhere. Likewise,
A({h > 1}) = f hdv > A({h > 1}) + m({h > 1}),
{h>1}
which implies that h < 1 m-almost everywhere and so v-almost everywhere.
Hence we can choose a representative of h such that 0 < h(x) < 1 for every
xEX.
Now let f be an m-integrable piecewise constant function. By (**), ($), and (t),
JidA JfhdA.
Tf = = Phdm+At
the same time, by approximating h with piecewise constant functions and using the continuity of T, we see easily that f f h dA = T(fh). We deduce that
Jfhdm.
T(f(1-h))=
Since this holds for every m-integrable piecewise constant f, it also holds for every f E LP(m) such that f > 0 (use an increasing approximating sequence; see the remark on page 146). Now let f E LP(m) be such that f > 0. For every integer k, inf(f/(1-h), k)E LP(m), so
T(inf(f, k(1-h))) = / inf(j f h, k)hdm.
By making k approach infinity and using again the continuity of T, we get
Tf =1 1fhhdm.
Thus, g = h/(1 - h) serves our purposes.
We now get, without having to assume that m has finite mass:
Lemma 2.3 If T E (LP)', there exists a measurable function g such that, for all f E LP,
fgEL' and Tf= /fgdm.
Proof. For f E L, set Tl f = Re(Tf) and T2f = Im(T f ). Then Ti and T2 belong to (LR)'. If Lemma 2.3 is true in the real case, we can apply it to Tl and T2 to obtain real functions g1 and g2, and clearly the function 9 = 91 + i92 works for T. Therefore we can suppose we are in the real case.
In this case T can be written as the difference of two continuous and positive linear forms on LPa (apply Remark 2 on page 88 to the lattice La).
So we can in fact suppose that T is a positive continuous linear form on Ls, and we do so.
Since the measure m is o-finite, there exists a countable partition (Kn) of X consisting of elements of 9 of finite measure. For each integer n, let mn be the restriction of m to K,,. If f E Lg(m,,), denote by f the extension of f to X taking the value 0 on X \ Kn. The linear form on Lg(m,,) defined by f H T(f) then satisfies on Kn the hypotheses of Lemma 2.2.
Therefore there is a positive measurable function g,, on Kn such that, for all f E LR (inn),
fgn E 4 (mn) and T(f) = Jig,, dmn