weak strong L , L andIntegration p p AppendixF

Appendix F

L _strong ^p , L _weak ^p and Integration

Order and simplification are the first steps toward mastery of a subject

— the actual enemy is the unknown.

— Thomas Mann (1875–1955)

For a function f : Q BB₂ , Bochner-measurability corresponds to the uniform (i.e., Banach space) topology of BB2 . However, for several applications it suffices that, e.g., f x : Q B₂ is measurable for each x B (i.e., that f is strongly measurable). We study this and the corresponding weak concept;

in particular, we define and study L_strong^p and L_weak^p spaces (in Section F.1; we note that L^∞_strong is usually a Banach space (Theorem F.1.9) but L_strong^p is often incomplete for p ∞(Example F.1.10)).

In Section F.2, we define and study integration and convolution for strongly or weakly measurable functions. In Section F.3, we treat H_strong^p and H_weak^p spaces and the Laplace transform of strongly or weakly measurable functions.

In this chapter, B, B₂ and B₃ denote Banach spaces over the scalar field K, where K R or K C (we have K C in Section F.3), U , H, and Y denote Hilbert spaces, and µ is a complete positive measure on a set Q.

997

F.1 L

and L

If you think you have the solution, the question was poorly phrased.

In this section, we study strong and weak (operator) measurability and corresponding L^pspaces. We start with the definitions of measurability:

Definition F.1.1 (Strong and weak measurability, LLL LLLstrong_strongstrong LLL_weakweakweak) By

LQ; we denote the (equivalence classes of) Bochner measurable functions

Q .

Let F : Q BB2 . Then F is strongly measurable (F L_strongQ; B B₂ ) if Fx LQ; B₂ for all x B, and F is weakly measurable (F L_weakQ; B B₂ ) ifΛF x L Q for all x B Λ B₂ .

Elements F G L (resp Lstrong, L_weak) are identified (FG, or F G, where F is the equivalence class of F) if F G a.e. (resp. Fx Gx a.e. for all x B,ΛF x ΛGx a.e. for all x B Λ B₂ ).

If F : Q BB2 and G : Q B₂ B are strongly (resp. weakly) measurable andFx Λ_B2

B₂

xGΛ_B

B

a.e. for all x B Λ B₂ , then G is the adjoint of F] in L_strong(resp. L_weak) and we write FG.

If F L Q;BB2 and G L Q; B2B3 , then we define GF :

GF L_strong Q; B B₃ .

The above definition can be generalized to situations where F is B₃-valued and there is a continuous bilinear mapping B B3 B2, but these situations can be reduced to the above by considering B₃as a subspace of BB₂ . The definition of G F will be justified in the proof of Lemma F.1.3(b).

We write FL, FLstrong or FL_weak when there may be confusion about the sense in which an equivalence class is defined. We use the standard convention to write F in place of F when there is no risk of confusion.

The Bochner measurability of an operator-valued function is often called uniform measurability (in the literature, also the term “strong measurability” is used, but we shall use that term for L_strongonly).

We shall interpret the definition of Lstrong and Lweak for vector-valued func- tions as follows: if f : Q B, then we consider f as a function Q^! K; B , so that strong (operator) measurability reduces to Bochner measurability and weak (operator) measurability reduces to “weak vector measurability”, i.e., to the con- dition thatΛf L for allΛ B (for operator-valued functions, by weak measur- ability we refer to weak operator measurability, as defined in Definition F.1.1).

Lemma F.1.2

(a) LQ; B B₂" L_strongQ; B B₂" L_weak Q;BB₂ .

(b1) If F^# L_strong Q;BB₂ has an adjoint in L_strong, then this adjoint is unique.

(b2) If FF^$ L_strong, then F^%_Lstrong ^&F_L

strong. If F F^$ L_weak, then

F_Lweak ^'F_L

weak.

(c) Let B be reflexive. If F L_weak Q;BB2 , then F⁽ L_weakQ;B₂ B and F^)'F^* .

In contrast to (a) and (b), in Example 3.1.4 we construct F L_strong+ L s.t.

F^, L_weak+ L_strong and ess sup^- F Λ^- B ∞ for certain Λ B₂ , even though F ^.0Lstrong and hence F_{L strong} ^/0_{L strong} ^/0Lstrong. Thus, e.g., F⁰ Lstrong

may have an adjoint in L_strongeven if F²¹ L_strong.

Proof: (a) This follows from (B.18) (Naturally, the inclusions should be injective. By Lemma B.2.6, this is the case (but the equivalence classes may be enlarged with “less measurable” elements and there may appear new equivalence classes as we move from L to L_strongor from L_strongto L_weak).)

(b2)&(c) These follow directly from the definition.

(b1) For F³ L_strongQ; B B₂ we have, by Lemma B.2.6, that F⁴

0⁶⁵ ΛFx 0 a.e. for all x B Λ B₂.

If F^*7 G and F^*8 H, then x G⁹ H Λ^: 0 a.e. for all Λ B₂ x B. Therefore, hence G⁹ H Λ 0 a.e. for all Λ B₂ , by Lemma B.2.6 (because ^; x B^<

< - x^- _B = 1^> "

B is norming), i.e., G^?H.

@

Now we go on with further properties of strongly and weakly measurable functions:

Lemma F.1.3 Let F⁰ L_strong Q;B B₂ , G^A L_strongQ;B₂ B₃ , f^A LQ; B , h^B LQ , H : Q B B2 and 1 ⁼ p ⁼ ∞. Then we have the following:

(a) F f LQB₂ and hF L_strongQ; B B₂ . (b) We have GF^# Lstrong

Q; B B3 . In particular, Lstrong

Q B is an algebra.

(c) If H_n L_strongfor all n N and H_n H a^Ce^C, then H L_strong. (d) If dim B ∞, then Lstrong

Q;B B2 L Q;BB2 (and FLstrong

F_L); if B₂ is separable, then L_weak Q;B B₂

L_strong Q; B B₂

(and F_Lstrong F_Lweak).

(e) If also R is a measure space, then qrED Fq

L_strongQ R and

q rFD Fr⁹ q L_strongQ R .

(f1) Assume that B is separable. Then ^- F^- is measurable and ^-GF^- _L^∞_strong ess sup^- F^-%HJI_B

B2^{K =} ∞, in particular, F^?0⁶⁵ F 0 a.e.

Moreover, F^4ML L_strong iff F q ML

for a.e. q Q and F^{N 14} L_strong. If, B is also reflexive, then F^O Lstrongand F^?F^* .

(f2) Assume that Q is separable and µΩ^AP 0 for openΩ^" Q.

Then^Q Q;BB2^" LQ; B B2^" Lstrong

Q; B B2 . Moreover,

Q b

Q;BB₂R" L^∞ Q;BB₂R" L^∞_strongQ; B B₂ , with equal

norms.

Assume, in addition, that F ^SQ Q;BB2 . Then F^%TUF and

-GF^V- _L^∞_strong sup_Q^- F^{- HJI}_B

B₂K = ∞; in particular, F^)'0^W5 F ^X 0.

If F q YL

B B₂ for a.e. q Q, then F ^{N 1 Z}F^{N 1FYL} L_strong. Con- versely, if F^[SL L_strong and G^#\F ^{N 1} and ^- G^{- HJI}_B2

BK = M a.e., then F q

]L

BB₂ for all q Q.

(g) Assume that B₃ B. Then any separable sets X₀ " B and Y₀ " B₂ are contained, respectively, in closed separable subspaces X " B and Y " B₂, s.t. there is a null set N " Q satisfying Fq x Y and Gq y X for all x X , y Y and q Q⁺ N. (Cf. Lemma 3.2.6.)

(w) Replace L_strong by L_weak everywhere above in this lemma. Then parts (c) and (e) above hold, we have hF L_weakQ; B B₂ , and g L Q; B₂

implies that gF f LQ . Parts (f1) and (f2) (provided that both B and B₂ are assumed to be separable in (f1)) also hold except possibly the claims concerning ^L L_strongand ^L L^∞_strong.

Moreover, if f L, then F f L_weak; if F L_strong, then GF L_weak; Note that any measurable subset of Rⁿ with the Lebesgue measure (or any countable set with the counting measure) satisfies the assumptions of (f2), by Lemma B.2.3(e).

Proof: (a) The claim on hF is a special case of (b). Now g_j :

∑_k^j^ 1Fx_kχ_Ek L Q; B2 for all j N, ^; x_k^{> "} B and disjoint, measurable E_k (k N). Therefore, F f lim_j_,` ∞g_j L, when f ∑^∞_k^ 1x_kχ_Ek. If f L is arbitrary, f_n f a.e., and f_nis countably-valued (n N), then L^a F f_n F f a.e., as n^!b ∞, hence then F f L, by Lemma B.2.5(c).

(b) 1^c Now Fx L, hence GFx L, by (a), for any x B. Thus, GF³ L_strong.

2^c We shall now show that G F :^\GF is well defined, as promised below Definition F.1.1: Let F^de(F, G^deRG and x B. By Lemma B.2.5(b1), there is a separable subset B₀ " B s.t. Fq x B₀for a.e. q Q. Choose a null set N " Q s.t. Fy F^dy on N^cfor all y in a dense, countable subset of B₀, hence for all y B₀. Then FGx F^dG^dx a.e. on N^c, hence a.e., hence FG^?fF^dG^dg, hence multiplication is well-defined.

3^c Apply 1^c to B₂ B B₃to see that L_strong Q;B is an algebra.

(c) Now Hx limnHnx a.e. hence Hx Lstrong, for any x B (thus, it were sufficient if H_n H strongly).

(d) We assume that dim B ∞ and prove that F LQ; B B₂ . Take a base e1^CCCh en ^" B, and set fj: Fej LQ; B2 for all j. Then F∑ⁿ_j^ 1αjej

∑ⁿj^{^} 1αjf_j, hence F ∑j f_jP_j L, where P_j B is the mapping∑ⁿj^{^} 1αje_j D

αj. Obviously, Fx 0 a.e. for all x B iff F 0 a.e., hence 0_Lstrong ^'0_L. If, instead B2 is separable, then any F Lweak

Q;B B2 is strongly measurable, by Lemma B.2.5(b1) (and ΛFx 0 a.e. for allΛ B₂ iff Fx 0 a.e., hence 0_Lstrong 0_Lweak).

(e) Now F^Vi x LQ B2^j" LQ RB2 for all x B, hence q^D Fq is in L_strongQ R . The second claim follows analogously, by using Lemma B.2.9.

(f1) 1^c Let ^; b_k^> _kk N be dense in the unit ball of B. Then ^- F^{- HJI}_B

B2^K

sup_k^- Fb_k^- _B2is measurable and ^- F^- _L^∞_strong sup_k^- Fb_k^- _L∞ = ess sup^- F^- . But if

- F^{- P} M on E with µE^3P 0, then^l _kE_k E, where E_k:^; q E^<

< - Fq b_k^{- P} M^> , hence then µE_kmP 0 for some k, thus ^- Fb_k^- _∞ P M. Consequently, sup_k^- Fb_k^- _L^∞ = ess sup^- F^- .

2^c Claim F ^{N 1 n}F^{N 1}: If F^{N 1} exists a.e. and F^{N 1o} L_strong, then, obviously, F^{N 1} F⁾ I and F F^{N 1)} I in L_strong.

For the converse, assume that B₃ B and G^4pF ^{N 1}. Set X₀: B and apply (g) to obtain a null set N " Q and a closed separable subspace Y " B₂s.t.

Fq B ^" Y for all q Q⁺ N.

Now I_B⁹ GF 0 a.e. on N^c and I_B2 ⁹ FGrq

Y 0 a.e. on N^c Therefore, Gq

Y F^{N 1}a.e. on N^c, say, on N₁^c, where N₁is a null set. But for any y B₂we have F Gy y a.e. on N₁^c, hence B₂ Y , i.e., G F^{N 1}on N₁^c, hence a.e.

3^c Assume now that B is also reflexive. LetΛ B₂ . Thenx F Λ^)M F xΛ is measurable for all x B B^h , and F Λ: Q B is separably-valued (B is separable, by Lemma A.3.4(R2)), hence F Λ is measurable, by Lemma B.2.5(b1). Obviously, F^%?F^* .

(f2) (In fact, piecewise continuity suffices (or that Q^sl _nk NQ_n, where, for each n, Qn ^" Q is a Borel set, F ^Q Qn;UY , and µ Ω^P 0 for all open

Ω ^" Q_n).) Note that we implicitly assumed that Q is a topological space and

that all Borel-sets are measurable.

1^c By Lemma B.2.5(e),^Q " L. (Note that we have identified F and F_Lfor F ^tQ ; by 3^c , this inclusion is injective.) Combine this with Lemma F.1.2(a) to obtain ^{Q "} L ^" Lstrong.

2^c We have ^- F^- _L^∞_strong sup_Q^- F^- : If F ^tQ and ^- Fq x^- P M :^'- F^- _L^∞_strong for some x B s.t. ^- x^- = 1, then Ω:^'; q Q^<

< - Fq x^- P M^> has a positive measure, hence ^- Fx^- _∞ P M, a contradiction, hence sup_Q^- F^{- HJI}_B

B2^{K =} M,

hence sup_Q^- F^{- HJI}_B

B₂K M.

3^{c Q} _b " L^∞" L^∞_strongwith equal norms: this follows from 1^c and 2^c .

4^c Now also F is continuous, hence F L " L_strong, by 1^c . From the definition of F we observe that F ^?'F .

5^c If Fq uL

B B₂ for q N^c, where N is a null set, then F^{N 1}

Q

N^c;B2 B^A" Lstrong

N^c;B2 B Lstrong

Q; B2B .

6^c Assume that F^0SL L_strong, G^3/F ^{N 1} and ^- G^{- HJI}_B2

BK = M on N^c₀, where N₀is a null set.

Let x₀ B and y₀ B₂ be arbitrary. Set X₀ :^\; x₀^> , Y₀ :^v; y₀^> , and apply (g) to obtain closed separable subspaces XY and a set N s.t. Fq X " Y and GqY " X for all q N^c, x₀ X " B and y₀ Y " B₂. By continuity,

Fq X " Y for all q Q.

Since GFx x and FGy y a.e. for all x B and y B₂, hence for all x X and y Y , we haveF ^{N 1w}G also in L^∞_strongQ;Y X . Thus, we can apply (f1) to obtain that Fqxq

X ^yL

XY for a.e. q Q, say for q N_x^c₀

y0, where N_x0

y0 is a null set.

We now show that Fq q

X ^ZL

B B₂ for all q Q: To obtain a contradiction, assume that Fq0 ^qX ^1RL

B B2 for some q0 Q. Then there is

an open V "

XY s.t. Fq0 V and T V & T ^zL XY[{

- T^{N 1-} P

M, by Lemma A.3.3(A4). It follows that V^d :^f; q Q^<

<

Fq q

X V^> is open and V^d " N₀^l N_x0

y0, hence µV^d 0, hence V^d6 /0, a contradiction.

In particular, F q x₀ ¹ 0 and y₀ RanF q , for all q Q. Because x and y were arbitrary, we have Ker Fq

|; 0^> and RanF q

B₂, hence Fq

]L

B B₂ for any q Q.

(g) Let DX ^" X0 and DY ^" Y0 be dense and countable. For any n N, we have FGF

n

GF

nG GF

n

FG

n

L_strong, by (b).

For each x D_X, there is a null set N₀^x" Q s.t. Y₀^x: FQ+ N₀^xx is separable.

Set Y₁: spanY₀^l}l _xk DXY₀^x , N₀^d :^sl _xk DXN₀^x. It follows that FQ+ N₀^d x " Y₁ for all x X₀, by continuity. Moreover, Y₁ " Y is separable, by Lemma B.2.3(a)&(c), and N₀^d is a null set.

For each k 1^b N, given N_k^d and Y_k, choose, analogously, a null set N_k` 1

and a separable subspace X_k` 1 <sup>"</sup> B s.t. X<sub>k</sub> " X<sub>k</sub>` 1 and GQ+ N_k` 1Y<sub>k</sub> " X<sub>k</sub>` 1. On the other hand, for each k 1^b N, given N_k and X_k, choose a null set N_k^d and a separable subspace Y_k " B₂s.t. Y_k

N

1 ^" Y_kand FQ+ N_k^dX_k " Y_k.

Set N :^sl _kN_k^l N_k^d, X : span ^l _kX_k , Y : span ^l _kY_k . If q Q⁺ N, then Fq x Y for all x ^~l _kX_k, hence for all x X , by linearity and continuity;

analogously, Gq y X for all y Y .

(w) 1^c F fGFhFgF f L_weak: Let f L. A slight modification of the proof of (a) shows that F f L_weak. Let now G L_weak and F L_strong. Then Fx L for each x B, hence GFx L_weak, by the above; consequently, GF L_weak. The claims on hF and gF f follow.

2^c The other claims: The above proofs of parts (c), (e) and (f) need only be slightly changed (in (f) we use a countable norming subset of B₂ and a dense

subset of B). ^@

Definition F.1.4 (LLL_{strongstrongstrong}^ppp Q;Q;Q;BBBBBB₂₂₂ ) Let 1= p= ∞.

By L_strong^p Q; B B₂ we denote the space of F^o L_strongQ; B B₂

having a finite norm

- F^- _L^p

strong : sup



x



B^€ 1

- Fx^- _L^pI_Q

B2^{K C} (F.1)

By L_weak^p Q; B B₂ ) we denote the space of F^j L_weakQ; B B₂

having a finite norm

-GF^V- _L^p

weak : sup



x



B

Λ^_B2^€ 1

- ΛFx^- _L^pI_Q

K C (F.2)

It follows that L_strong^p Q;BB2 L^p Q;BB2^E

L^pQ; B2 n

when n : dim B ∞ (cf. Lemma A.1.1(a4)). Note also that ^- F^- _L^p

weak ⁼

- F^- _L^p

strong ⁼

- F^- _L^p for F L^pand that ^- F^- _L^p

strong (resp. ^- F^- _L^p

weak) is the norm of the operator B ^a xD Fx L^p(resp. the “bilinear norm” of B B₂ ^a x Λ^AD ΛFx L^pQ ; cf. Lemma A.3.4(J1)).

One could insist that L_strongQ; B B₂ should consist of all linear F : BD

L_strongQ; B₂ (and analogously for L_weak, L_strong^p , L_weak^p ), not just for those that

take the form of a function (a.e.). See Theorem F.2.1(g) etc. for details. However, that broader definition would cause problems in several applications.

The spaces L_strong^p and L_weak^p are normed spaces:

Lemma F.1.5 Let 1 = p= ∞. Then

(a1) L_strong^p Q;BB₂ is a subspace of BL^p Q; B₂ with same norm.

(a2) L_weak^p Q;BB2 is a subspace of B B2 L^pQ with same norm.

(b) If Q " Rⁿ and µ m, and B is a Hilbert space or B₂ K, then L^∞_strongQ; B B2

BL^∞Q; B2 .

(c1) We have L^pQ; B B₂‚" L_strong^p Q;BB₂ƒ" L_weak^p Q; B B₂ , continuously.

(c2) If F L_strong Q;B B₂ , then ^- F^- _L^∞_strong ^- F^- _L^∞_weak; if F L Q;BB₂ , then ^- F^- _L∞ ^|- F^- _L^∞

weak.

(d) If F L_strong^p Q; B B₂ and T B₂ B₃ , then T F L_strong^p Q; B B₃ and ^- T F^- _L^p

strong ⁼

- T^{- H -} F^- _L^p

strong. Also the analo- gous “weak” claim holds.

(e) If F L_weak^p Q; B B₂ and B is reflexive, then F^E L_weak^p Q; B₂ B

and ^- F^„- _L^p

weak

…- F^- _L^p

weak.

(f) (dim Bdim Bdim B ∞∞∞) If dim B ∞, then L^p Q; B B₂

L^p_strong Q; B B₂

(with equivalent norms). If dim B₂ ∞, then L_strong^p Q; B B₂

L_weak^p Q;BB₂ (with equivalent norms).

(g1) Assume that p ∞or µ isσ-finite. Then Hg L for all g L^p Q; B iff H L_strong.

(g2) Assume that p ∞or µ is non-atomic. Then Hⁱ L^p iff H L^∞_strong.

“Usually” L_strong^p and L_weak^p are Banach spaces only for p ∞; see Theorem F.1.9 and Example F.1.10 for details.

Proof: (a1)&(a2) These are obvious.

(b) Let F B L^∞Q; B2 . W.l.o.g. we assume that Q Rⁿ (replace F by Fχ_Q). Set M :^'- F^{- H} . For any q Rⁿ, the set X_q:^S; x B^<

<

q Leb LFx >

is a subspace of B, and ^- LFx^- = - Fx^- _∞ = M^- x^- _B on Q (x B), by Lemma B.5.3.

For each q Q, the map x D LFxq is obviously linear on X_q, hence it has a norm-preserving extension G q

BB₂ , by Lemma A.3.11, so that

- Gq ^{- HJI}_B

B2^{K =} M.

Let x B. Then for a.e. q Q we have x X_qand hence LFq x G q x;

but LFx Fx a.e., hence Fx Gx a.e. Consequently, G : Q B B₂ is strongly measurable and ^- F ⁹ G^{- HJI}_B

L^∞IQ;B2^K†K 0.

Thus, we have constructed G L^∞_strongQ; B B₂ s.t. G F as an element of B L^∞Q; B₂ . Finally, G satisfies the additional condition

- Gq ^{- HJI}_B

B2^{K = -} G^- L^∞_strong for every q Q.

(c1)&(d) These are obvious (and the norms of the embeddings in (c1) are at most one).