On the Representation theorems of Riesz and Schwartz

Transcription

1 On the Representation theorems of Riesz and Schwartz Yves Hpperts Michel Willem Abstract We give simple proofs of the representation theorems of Riesz and Schwartz. 1 Introdction According to J.D. Gray in [2] Only rarely does the mathematical commnity pay a theorem the accolade of transforming it into a tatology. The Riesz representation theorem has received this accolade. Indeed for Borbaki (see [1]) a measre is a continos linear fnctional. In most textbooks, the Riesz representation theorem asserts that continos linear fnctionals on C(K), where K is a Hasdorff compact space, are represented by integrals with respect to a reglar Borel measre. It is interesting to note that many variants of Lebesge integral were defined by. Riesz in order to avoid the se of measre theory! Bt the original theorem in [4] asserts that continos linear fnctionals on C([, 1]) are Stieltjes integrals : Etant donné l opérater A(f(x)), on pet déterminer la fonction à variation bornée α(x), telle qe, qelle qe soit la fonction contine f(x), on ait 1 A(f(x)) = f(x) dα(x). Received by the editors October 24. Commnicated by J. Mawhin. Key words and phrases : measre theory, distribtion theory, Stieltjes integral, bonded variation, representation theorem. Bll. Belg. Math. Soc. 13 (26),

2 436 Y. Hpperts M. Willem Ths the original Riesz representation theorem gives a correspondance between continos linear fnctionals and fnctions of bonded variation. Moreover, as observed by Horváth in [3], since, for f reglar with compact spport in ], 1[, 1 f(x) dα(x) = 1 f (x) α(x) dx, the fnctional A is the derivative, in the sense of distribtions, of α. In this note we give a short elementary proof of the original Riesz representation theorem and of its extension in two dimensions in [3]. Contrary to [3], we se only two tools : Abel transform and the fact that a bonded fnction (on ], 1[ or ], 1[ ], 1[) is Riemann integrable if and only if it is almost everywhere continos. 2 The one-dimensional case We recall some basic definitions. We call K(], 1[) the space of continos fnctions from ], 1[ to R with compact spport in ], 1[ and D(], 1[) the sbset of infinitely derivable fnctions of K(], 1[). Definition 2.1. The space of bonded measres on ],1[ is defined by (K(], 1[)) := {µ : K(], 1[) R; µ is linear and there exists C > : 1 dµ C for all K(], 1[)}. Definition 2.2. A fnction L 1 loc(], 1[) is of bonded variation if { 1 } D = D ],1[ = sp (t) (t) dt; D(], 1[), 1 <. After these preparations, we can state the representation theorem of. Riesz in the following form : Theorem 2.3. Let µ be a bonded measre on ], 1[. Then, there exists a fnction :], 1[ R of bonded variation sch that, in the sense of distribtions, µ =. Moreover, µ = D. Proof. Let (t) = µ(], t]). By contable additivity of the measre µ, is continos otside a contable set. Let s take D(], 1[) and define n = 1 ( ) (] ]) k 2 χ k, k+1 n 2. It is clear n that n niformly on [, 1]. Using Abel transform, we have 1 dµ = lim n = lim n 1 n dµ ( ) (] k k µ 2, k + 1 ]) n ( ) [ ( ) ( )] k k + 1 k 2 1 = lim n 2 = lim n 2 1 = lim n ( ) [ k ( k ) ( ) k ( k θk ( )] k 1 ),

3 On the Representation theorems of Riesz and Schwartz 437 where < θ k < 1. or every ε >, there exists δ > sch that x y δ (x) (y) ε. Hence 1 lim n dµ 1 2 ( ) ( ) k k2 ε n. Since is Riemann integrable on ], 1[, it follows that 1 inally we obtain 2 1 dµ = n lim { 1 D = sp = sp { 1 ( ) ( ) k k2 1 = (t) (t)dt. n } (t) (t) dt; D(], 1[), 1 } dµ; D(], 1[), 1 = µ, and the proof is complete. Corollary 2.4 (Schwartz representation theorem). Let f be a distribtion on ], 1[ sch that f is a bonded measre µ on ], 1[. Then f is a fnction of bonded variation. Proof. Let s define the fnction (x) = µ(], x]). Then (f ) = µ µ and f = c R.

4 438 Y. Hpperts M. Willem 3 The higher-dimensional case In order to be readable, we shall consider the two dimensional case. Definition 3.1. The space of bonded measres on =], 1[ ], 1[ is defined by (K()) := {µ : K() R; µ is linear and there exists C > : dµ C for all K()}. Lemma 3.2. Let D(). or ε > there exists < δ < 1 sch that (x + h, y + k) (x, y + k) (x + h, y) + (x, y) x y (x, y)hk ε h k for all (x, y) provided that h, k δ. The following lemma is contained in [3]. completeness. We give the proof for the sake of Lemma 3.3. The fnction (x, y) = µ(], x] ], y]) is continos otside a Lebesgenegligeable set Proof. Since µ is the difference of two positive measres, we can assme that µ is positive. Since µ( ) < +, by contable additivity of the measre µ, only a contable nmber of horizontal or vertical segments in can have a strictly positive µ-measre. Let H be the collection of all horizontal segments σ, going from the left bondary to the right bondary of sch that µ(σ) >.Then H is contable. A similar assertion holds for the collection V of vertical segments σ sch that µ(σ) >. Denote by S the set of all points of which lie on a segment belonging to H or to V. Then S has the Lebesge measre zero. We claim now that is continos at every point (x, y) not belonging to S. We have ], x + h] ], y + h] =], x] ], y] and h> ], x h] ], y h] =], x[ ], y[. h> By the choice of (x, y) we have µ({(x, η); < η y}) = µ({(ξ, y); < ξ x}) =, and lim (x + h, y + h) = lim µ(], x + h] ], y + h]) = µ(], x] ], y]) = (x, y). h h or x h ξ x + h and y h η x + h the vale f(ξ, η) lies between f(x h, y h) and f(x + h, y + h), from which the lemma follows. Theorem 3.4. Let µ be a bonded measre on. Then there exists a fnction L () sch that, in the sense of distribtions, µ = x y.

5 On the Representation theorems of Riesz and Schwartz 439 Proof. Let D(). Let s define n := 1 ( ) (] ]) k l, 2 χ k, k+1 n χ (] ]) l, l+1 2. It is clear that n n niformly on. Using Abel transform we have dµ = n lim = lim 2 n + = lim 1 n + = lim 1 n + By lemma 3.2 we have, for every ε >, lim n n dµ ( ) (] k µ n 2, k + 1 ] ] l n 2, l + 1 ]) n ( ) [ ( k + 1 n 2, l + 1 ) n ( ) ( )] k n 2, l n ( ) [ ( ) ( k n n 2, l 1 ) n ( ) ( k n 2, l 1 )]. n dµ n ( k 2, l + 1 n ( ) ( ) k n x y 2, l n ε. By lemma 3.3, x y is Riemann integrable on. Then we obtain dµ = 2 1 ( ) ( ) 1 k lim n + 2 2n n x y 2, l n = (x, y) x y (x, y) dxdy. Remark 3.5. Let (ρ n ) be a Dirac seqence : ρ n D(R 2 ), spp ρ n B[, 1/n], R ρ dxdy = 1, ρ n. Then for every K(), dµ = lim x y (ρ n ) dxdy. n or a proof see [5] p. 57. )

6 44 Y. Hpperts M. Willem References [1] G. CHOQUET, L analyse et Borbaki, Enseign. Math. II. Sér. 8, pp , [2] J.D. GRAY, The shaping of the Riesz representation theorem : A chapter in the history of analysis, Arch. Hist. Exact Sci. 31, pp , [3] J. HORVATH, A Remark on the Representation Theorems of rederick Riesz and Larent Schwartz, Mathematical analysis and applications, Part A, pp , Adv. in Math. Sppl. Std., 7a, Academic Press, New York-London, [4]. RIESZ, Sr les opérations fonctionnelles linéaires, C. R. 149, pp , 199. [5] M. WILLEM, Analyse onctionnelle Elémentaire, Cassini, Paris 23. Université Catholiqe de Lovain, Institt de mathématiqe pre et appliqée, Chemin d Cyclotron 2, B-1348 Lovain-la-Neve, Belgiqe. hpperts@math.cl.ac.be willem@math.cl.ac.be