A GEOMETRC ROOF FOR SOflE SMLE CASES OF :iartt!gale CONVERGENCE BRALEY SMS The use of real valued martigales i probability theory has grov steadily sice their detailed itroductio by Doob i 195. Vector valued martigales vere first studied explicitly i 196 by Chatterji ad idepedetly Scalora, ad vere rapidly recogised as a pwerful tool for the study of certai Baach space structures (see, for example, Diestel ad Uhl, 1977). Besides itroducig the idea of a vector valued aartigale this article gives a "geometric" proof of the followig covergece theorem which makes o use of probability theory ad ivolves a miim~ of measure theory. 'HU)RM 1. Let (,, fl be a martigale i L (,u;x/ uhere 1 < p < - m.d X is a uifom!? muex Bmroch space. The :here eaiste fw c L (Z,g;X) 1~6th lf + if (ad a3 if) 1 i\/? 5 K for all d.ere X >. Our proof is particularly simple whe p = ad X is a Hilbert space. Sice this case vould usually be itroduced first, the proof for other cases has bee relegated to foototes ad remarks. The article oves much to the suggestios of my colleagues, i particular. Xr. L. Kavalieris, made durig a series of semiars at the Uiversity of New Eglad. 5 1 RELMNARES. Throughout the article, 1 deotes a o-algebra of subsete of Ci ad p a coutably additive probability measure o Z (that is p(el for all E c Z ad ~(') = 1). R is a Hilbert space over the real field, vith ier product (-,-, ad X is a real Baach space. We say X ie uifomly cavez if: every sequece (z-l for which Z + Z r 1 have a comro limit as,m +-, is a Cauchy sequece (ad hece llz 11 ad 1i coverget).() t is a easy cosequece of the parallelogram rule that B is uifordy - Z Z m +Z covex (11-1 = YZ~ + YZ~~ - ~ ~ ). () Bocher itegmtiu for vector valued fuctiue. The reader iterested oly i real valued martigales could pass over this material provided B (the real umbers) is everywhere substituted for 6' or X ad itegrals (1) This form of the defiitio is most suited for our purpose. The equivalece with Clarkso's origial defiitio beig a simple matter. () For details see either Duford ad Schvartq1958 or Diestel ad Uhl, 1977.
are Fterpreted i the sese of Lebesgue. A fuctio j: R -+ X is u-measurcbze Ff there exists a sequece of sfmple fuctios (S with S (wl - f(wlll * for u-almost all w E R. A simple fuctio is of the form e 1, which,without loss of geerality, may be N assumed to be pairvise disjoit. Here wd elsevhere, X is the cl-aracteristic fuctio E of rhe set E. The itegral of such a slmple fuctio f is defhed i the obvious vay; for some z1,xz,..., xn e X ad El,..., E A u-measurable fuctio is itegrable if there exists a sequece S) of s'sple fuctios with R - f d *. (Lebesgue itegral) this case the orm lfmit, as * m, of S d~ exists for each E E Z ad is ide- pedet of ay particular choice for S. By defiitio, f du is this commo limit. L 'C For 1 S p < we deote by L Z,u;X) the space of f: R * X for which 1 = J' ilf(u)li E~)B r R < o. t is a sigificat result that the elemets of L (1,u;X are precisely those 1 i.aciioe which are itegrable i the above sese. t follws from K61der1s iequality d the filteess of u that the elemets of L fz,p,x) are itegrable. Further, as i the scalar case, 1. is a Baach space orm for this space. 13) 1-3. LZZ,u;lil is a Hilbert space. Fim. = j(fw),gu)) dp defier a ier-prod,wt o LZ(Z.~;%) X.3. Day r19411 proves the followig. A 3. L Z,p;X) is AfomZy mvez if 1 < p < - - X is 9orifsmLy cvez. p.71 Strictly, th space of equivalece chses modulo almost everyvhere zero fuctios. - 5 -
Let Z be a proper sub -algebra of 1. the L (Z u;x) is a proper subspace oj of L Z,u;X). i Clearly for ay give f e L (E,u;X) there ca be at most oe f 6 L (1 p;x) r o' ( with je f & = b fo dy for all E e Lo. Whe such a f exists it will be referred to as the -it-kmal eqecidtia 3f f with respect to Z ad deoted by E(f(ZO). L W 4. fo r Mo = L (Z,u;H) i the wditioal erpectatia of f :Ath respect to Z, df cd mzy if (f- fo,g) = for all g e Mo.??,w?. t is sufficiet to cosider g = de where z e A ad E e ZO (Mo is the closed liear spa of such fuctios). Nw, >:a if - fg,z XS = if ad oly if (1 if - fo)iui du,z) = for all r r a. E - While it is ot ecessary for our subsequet work, oe kportat cosequece sf ixs lemma is the existece of E(f 1 ~ for ~ ) every f e L(ZJp;B) a& sub a-algebra ZO. T-is follws sice fo is the foot of the per~edicular frm f' to H if ad oly if fo is :he closest poit frw No to f, the existece of which is emured by the uiform ccr~exity of L (Z,u;B!. Aother cosequece is the follwig. C:XLUKY 5. For f r LiZJy;A), E(flZG)llZ 5 llfl1. ( ) ~rzieed rip -cpig j + ~ ( iol f le z ~ar me lie projectio mto L(Zlr~;B. -- - : LCF. 1: C 11 = ifo,,fo) = (f - if - fo),fo) = (f,fo), by lemma 4. 5 \~lillf 11, by the Cauchy-Schvartz iequality. S
By a.mtigaze fz f i L fz,p;x we mea a ested sequece ' 5... ; Z of sub o-algebras ad a sequece of fuctios T, 5 Li S... d - - f,... with f 6 L fz,b;x, -p'fl,..., f- = 7:fE 1 Z j for a11 m. m (51 which satisfy the mrtirqale corditio As a example ote that for ay ested sequeces of sub a-algebras Z,, 5... 5 E 5... r Z ad ay f c LfZ,p;H, (1,E(fjEj) is a aartigale. To check the martigale coditio observe that for r m ad E c Zm we have J E EfflZm) du = f du, by the defiitio of -f. (., JE = EfflZ d,~, agai by the defiitio of E(-(. ad the JE fact that E r..?yc,~ of t& Mai Theorem. Ue are ow i a positio to prove Theorem 1..;/ A proof of this result for geeral p c [?,- is the follwig. By their desity we may assume that f is a shple fuctio, the r Jr: Eff Z h.11 XE 1 (1 du, by Jese's iequality. G- 1 t t = 1, f lzill X ) dp, by defiitio of.ei. /zo) ad the fact i=? Ei il c r. 't, Settig ufe = je f dp, for a11 E E Z, the m.artig.de coditio may be re-expressed as: For m the restrictio of u to Z is pm.
, Let ('i f l be a martigale i L(Z,p;H) with lf U 5 K for all. Let = L,!E,;i;Hl, the we have the follovig structure.. - A ested sequece of closed subspaces M < M 5... s M 5... ad a szifcrmly bouded sequece fo,fl,...,fj... - with f e M. By the martigale coditio ad corollary 5 we have llfuz 5 forsm.?"us.soll, ]fl 11,...,llfll,... is a icreasig sequece of real ders bouded aowe by K, ad so coverget from below to some real umber. Further, for < m r ile(f + ~;lr/z~lll~ = ~ l l f 1 lagai, by corollary 5 ad the martigale coditio), so + -- ' 'm li- 11 -+ k as m, -t -.?ece, by the uiform covexity of L (1, p;h'), fo, fl,...,f,... is a Cauchy sequece zr;d sc by the completeess of the space, coverges to some f- e LiZ,;?/. - f Z_ deotes the o-algebra geerated by u, it is trivial to check that? -- t J.4 = L(Z, p;h) - the proper home for our martigzle limit. - O the basis of our earlier observatios, the proof for arbitrary? E a d uiformly covex Baach space X is merely a paraphrase of the above. (,-) REFERENCES ;11 X.X. Day, "Some more uiformly covex spaces", Bull. her. Math. Soc. 9 (1941) - 54-57. 13 J. Diestel ad J.J. Uhl Jr., "Vector measures", ilmer. Hcth. Soc., Xath. Survey No. 15 (1977). C31 N. Duford ad J.T. Schvartz, Liem @emtars, art. tersciece. N.Y. (1956). THE LARGEST RME Readers of the h a Ageles Times (November 16, 1978) vill o doubt kziov cf che exploits of two freshme studets of CSC Hayvard, Laura Hichel ad Curt Soil, L ~ O, for a high school computig project, shoved that 371-1 (5553 digits) is prie. holl cotiued this work ad i February 1979 foud that '"" 793-1 (f567 digits) is ~rhe. A more recet aoucemet of the L.A. Times (Xay 31, 1979) reveals that the is :4497 tbe ext Hersere prime - (;3:35 digits) a result due tc rry tielso d T'avid Sluwisiii. - 55 -