Doob, The Development of Rigor in Mathematical Probability (1900-1950).pdf

The Development of Rigor in Mathematical Probability (1900-1950) Author(s): Joseph L. Doob Source: The American Mathematical Monthly, Vol. 103, No. 7 (Aug. - Sep., 1996), pp. 586-595 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2974673 Accessed: 07-11-2015 11:14 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 213.233.188.210 on Sat, 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions and nothingmore will everbe accomplishedby theseinventions. and some still place mathematicalprobabilityoutside analysis.todayone inventsthem in orderto make incorrectthe reasoningof ourfathers.YorkUniversity.the developmentis slow. edited 586 THE EVOLUTION OF . The developmentof science is not a simpleprogressionfrom one advanceto the next.In the 1930'sBanach spaces were sneered at as absurdlyabstract. with many wrong turns and blind alleys. from Deuelopmentof Mathematics by Jean-Paul Pier. . an idealization and analysisof certain real life phenomenaoutside mathematics. and now it is the turn of nonstandardanalysis.-Sept.188.. The following quotations(in translation)are relevant.In particular. Hermite: (in a letter to Stieltjes)I recoilwithdismayand horrorat thislamentable plagueof functionswhichdo not hauederivatives. This content downloaded from 213.OntarioM3J1P3.The mathematizationof probabilityrequired new ideas. .This paper is a brief informal outline of the history of the introductionof rigourinto mathematicalprobabilityin the first half of this century. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions .many statisticians and probabilistsresented the mathematizationof probabilityby measure theory. mathematicalprobabilitybecame a normal part of mathematics. In Reprinted with kind permission of Birkhauser Verlag AG. proceeds in a zigzag course. but ratherbecauseits opponentseventuaGy and a newgenerationgrowsup withit. The Developmentof Rigor in Mathematical Probability(1900-1950) Joseph L. 1994. Specificresultsare mentionedonly in so far as they are importantin the historyof the logical developmentof mathematicalprobability. and in particularrequireda new approachto the idea of acceptabilityof a function. Poincare: Formerly.when one inventeda new function. and remained for a long time.210 on Sat. Judged by hindsight. [Aug.in the first half of this centuiy. and full acceptanceof mathematicalprobabilit was not realized until the second half of the century. Edited by:Abe Shenitzer NorthYork.later it was the turn of locally convex spaces. Canada Mathematics. Mathematiciansare no more eager than other humansto embracenew ideas. Basel..THE EVOLUTIONOF. A new scientifictruthdoes not triumphby convincingits opponentsand die. Doob 1 Introduction. it was to furthersome practicalpurpose. ISBN 0-8176-2821-5 1900-1950.but gradually. and frequently moves in directions condemnedby leading scientists. makingthemsee the light. Planck: Probabilitytheory began.233. there are such simple contexts as the following. . This content downloaded from 213. adopted by a prominent Bayesian statistician. A more commonequallysatisfactorysolution is to leave fuzzy the question of whether the context under discussion is or is not mathematics.The simplestsolution. The following statements have been made about this law (my emphasis): Laplace: (1814) This theorem.view of the above quotationsit is not surprisingthat acceptanceof this mathematization was slow and faced resistance. seems to clusternear. The most obvious difficultyis that in the real worldonly finitelymanyexperimentscan be performedin finite time.an arguablequestion. is the vacuous one: never discuss what happenswhen a coin is tossed.what strikesone at once is what has been christenedthe law of large numbers. 3 The law of large numbers. In a repetitivescheme of independenttrials.has split into two quite differentcharms:those of real world probabilityand of mathematicalprecision.In fact even now some probabilistsfear that mathematizationhas removedthe intrinsiccharmfrom their subject. Yet the fact is that anyone tossing a coin observes that for a modestnumberof coin tosses the numberof heads in n tosses dividedby n seems to be getting closer to 1/2 as n increases.Perhapsthe fact that the assertionis called a law is an example of this fuzziness.that is. However.in which the law of large numbersis formulatedas a mathematicaltheorem. On the other hand. was difficultto prove by analysis. The moral is that the specific context must be examined closely before any probabilisticstatement is made. Anyonewho tries to explainto studentswhat happenswhen a coin is tossed mumbleswordslike in the long run. if the law of large numbersis to be stated in a real world nonmathematicalcontext.Suppose an individualrides his bicycle to work. when the bicycleis parked.And they are based rightin the sense that the charmof the old. In fact the relation between real world probabilityand mathematicalprobabilityhas been simultaneouslythe bane of and inspirationfor the developmentof mathematicalprobability. on nonmathematicaldefinitions.In the simple context of coin tossing it states that in some sense the numberof heads in n tosses dividedby n has limit 1/2 as the numberof tosses increases. 1996] THE EVOLUTION OF . or (coin context) if the coin starts close to the coin landing place and the initial rotational velocity of the coin is low. 2 VVhatis the real world (nonmathematical)problem? What is usually called (real world) probabilityarises in many contexts. . of insurance.it is clear that (tire context)if the ride is very short. in a desperate attempt to give form to a cloudy concept.impliedby commonsense. just as surprised as if 10 successivetosses of a coin all gave heads. it is not at all clear that the limit concept can be formulatedin a reasonableway.of statisticalphysics. the surprise would decrease and the probabilitycontextwould become suspect.which has never even had a universallyacceptable definition. vague probability-mathematics. if there is a mathematicalmodel for coin tossing.the valve on the front tire appearedin the upper half of the tire circle 10 successive days.it must be augmentedby an examinationof the physicalcontext. The riderwould be surprisedif. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions 587 .such as coin tossing.188.210 on Sat.233.But it must be stressed that manyof the most essentialresultsof mathematicalprobabilityhave been suggested by the nonmathematicalcontext of real world probability. If the law of large numbersis a mathematicaltheorem.The key words here are in some sense. tends. and so on. If philosophyis relevant. either the theorem is true in one of the variousmathematicallimit concepts or it is not. Besides the obvious contexts of gamblinggames. consistentwith conditionS.Its fundamentalconceptsare incompletelydetermined. Poincare: (1912) One can scarcelygive a satisfactorydefinitionof probability.genetics. .188.-Sept. there are n mutuallyexclusive. Mises: (1919) In fact. ratherthan philosophicalremarks. 588 THE EVOLUTION OF . This content downloaded from 213.have felt forced to think about the followingquestion.210 on Sat.unfortunately. except for the following remark on coin tossing.233. .Mathematicians. and m arefavorableto the eventA.such a project wouldbejust a tourde force. Mazurkiewicz: (1915) The theoryof probabilityis not an independentelementof mathematicalinstruction. v. insuranceand statisticalphysicsare here to stay. one can at least safely stateit. a solid body falls under the influence of gravity. then the mathematicalprobabilityof A is definedas m/n.. Pearson: (1935) (Oral communication)Probabilityis so linkedwithstatistics that.) E. but these conditions can show what the <<equallikelihoods>> depends on and therebygive it a plausibleinterpretation.can explainthe quantitativeinfluenceand importanceof the initial and final conditionsof the coin motion in order to justify allusionsto equal likelihood of heads and tails. [Aug.Its motionis determinedin Newton'smodel by his laws. If.or at least to write about it. but as it is impossibleto prove experimentally that it is false. These statements illustrate the enduring charm of discussions of real world probability.Ville: Bauer: (1939) One sees no reasonfor this propositionto be true. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions . . Only mathematicalprobabilitywill be discussedbelow.(He proceeded to make it into a mathematicaldisciplineby basing mathematical probabilityon a sequence of observations(<<Beobachtungen>>) with propertiesthat cannotbe satisfiedby a mathematicallywell defined sequence.Onlythese laws.. The foregoing should make obvious the advisabilityof separatingmathematical probabilitytheory from its real world applications.Gambling.althoughit is possibleto teachthe two separately.nevertheless it is verydesirablethata mathematicianknowsits generalprinciples. 4 What is probability? Here are some attemptsto answerthis question and to discussthe teachingof the subject. and any discussion of what the coin does cannotbe completeunless these laws are applied. Of course these laws can at best reduce the analysisto considerationsof the initial and final conditionsof the toss. one can scarcelycharac-terize the presentstate other thanthatprobabilityis not a mathematicaldiscipline. Newtonian mechanics provides a partial mathematical model for coin tossing. Uspensky: (1937) In a useful textbook. (translatedfrom the context of dice to that of coins) It is an experimentally establishedfact that the quotient.Theycontainmanyunsolveddifficulties.he gave the following once common textbook definition.Note however that no one doubts the real world applicabilityof mathematicalprobability.and equallylikelycases. In coin tossing. exhibitsa deviationfrom 1/2 whichapproaches0 for largen. In a lightermood he is said to have defined probability as a numberbetween 0 and 1 aboutwhich nothingelse is known. . If x1. The standardloose languagewill be used here. Recall that a Borelfield(-(r al.1) is the probabilitythat the sequence x hits both A and B. In particular(RN. The superscriptwill be omitted when N = 1. in more colorful language. Radon (1913) made the furtherstep 1996] THE EVOLUTION OF . It was alwaysclear that. a probabilityat least equal to.It was realizedthat contextsinvolvingaveragesled to probability. The point is that the left side of (5.the coupled ff algebramakingit into a measurable space will alwaysbe the cr algebraof its Borel sets.There were manyimportantadvancesin mathematicalprobabilitybefore 1900.Additivefunctionsof sets were of coursefamiliarto mathematiciansfrom conceptsof volume.and if A is a set of numbers.differenceequationsand differentialequations.considerthe probabilitythat an orbit of this motion throughpoints of a line hits A. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions 589 . The usual calculation(ignoringhere all notions of rigour). how to define a mathematical structurein which the variouscontextscould be placed.because measuretheory. the probabilitythat the sequence hits A n B. In fact + satisfies the 5 Mathematicalprobabilitybefore the era of precise defilnitions. which extended the definitionof volume in RN to the Borel sets.It was frequentlyclear how to use the contextsto suggestproblemsin analysis. howeverclassical mathematicalprobabilitywas to be developed. Measure theory started with Lebesgue's thesis (1902). long before 1900.but it was not clear how to formulatean overall mathematicalcontext. 6 The developmentof measuretheory. there was a minimumof attentionpaid to the mathematicalbasis of the contexts a maximumof attention to the pure mathematicsproblems they suggested. (S. that is.188. S).function A +(A) which in general is not additive.233. In the following. +(A n B). the strongsubadditivityinequality.1).developed in great detail in the second half of the century with the help of Choquet'stheory of mathematicalcapacity.S1) into a measurablespace tS2. .Although nonmathematicalprobabilisticcontexts suggested problemsin combinatorics. . are numbersobtainedby chance.1) whereasadditivityof + would implyequalityin (5. A measurablespace is a pair.but the subject was not yet mathematics.defines a inequality (t(A) + (h(B)-+(A U B) 2 +(A n B). or. A weaker condition than additivitywas less familiar but turned out to be essential later. S2) iS a function from S1 into S2 with the property that the inverseimage of a set in 52 is a set in S1. and in general greater than. . The sets of S are the measurable sets of the space.considerthe probability that at least one of the membersof this sequence lies in A. (5.210 on Sat. A measurablefunction from a measurablespace (S1.if S is metric.1).is satisfiedalso by the electrostaticcapacityof a body in R3. the concept of additivityof probabilityas applied to incompatiblereal worldeventswas fundamental.RN) denotes N dimensionalEuclideanspace coupledwith its Borel sets. . and this fact hints at the close connection between potential theory and probability. This unequaltreatmentwas inevitable.The class of Borelsets of a metric space is the smallest set (r algebracontainingthe open sets of the space. where S is a space and s is a (r algebraof subsetsof S.needed for mathematical modelingof real world probabilisticcontexts.kebra)of subsetsof a space is a collectionof subsetswhich is closed under the operationsof complementationand the formation of countable unions and intersections. The inequality(5.mass and so on. x2.had not yet been invented. This content downloaded from 213. 13 years after Lebesgue'sthesis. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions .As noted below. P). these digits are functions of x..210 on Sat.or samplepath. . or sample sequence if o is a sequence. Thus a stochastic process defines a function of two from S into S' is the tth randomvariableof the process.and such set functionswill not be discussedfurtherhere.188. pointed out that all that the usual definitionsand operationsof measuretheoryrequireis a Cr algebra of subsets of an abstract space on which a measure. or. is the statespace of the randomvariable. X. ').these functionsmiraculouslybecome random o 590 THE EVOLUTION OF . if the question is to be meaningful. and if the interval [0.A stochasticprocessis a familyof randomvariables{x(t. t E >} from some probabilityspace (S.) to general measuresof Borel sets of RN (finite on compactsets). P). X.Mutualindependence of randomvariablesis defined in the classicalway. Measure theory was developedas a part of classicalanalysis. X. into a state space (S'. where (S. the Radon-Nikodymtheorem (1930).). s).the function x(. for example to the (Lebesgue measure) almost eveiywhere derivabilityof a monotone function. by completion Finally Frechet (1915).In fact there may well be real world contextsfor which the appropriatemathematicalmodel is based on finitelybut not countablyadditiveset functions. a probabilitymeasureon this interval. '). fromOxS into the state space. At each step of this progressionnot necessarilypositive countablyadditive set functions-signed measures were incorporatedinto the theory. P). which gives conditionsnecessaryand sufficientthat a countablyadditivefunction of sets can be expressed as an integral over the sets.233. These measures are usually extended to slightly larger classes than the class of Borel sets.and applicationsin analysiswere immediate. [Aug. This content downloaded from 213. Borel (1909) pointed out that in the dyadic representationx = xtx2 . Thefunction x(t. S) is a measurablespace and P is a measure on s with P(S) = 1. 7 Early applicationsof explicitmeasuretheoryto probability. The set o is the index set of the process. (t.A probabilityspace is a triple (S. is defined. in which each digit Xj is either 0 or 1. when one writes carefully. The distributionof a random variablex is the measurePx on ' defined by setting PX(X)=P{SES X(S)Xt}@ The joint distributionof finitely many random variables defined on the same probabilityspace is obtained by making x into a vector and specifying' and S' correspondingly. s) from into S' is the sth samplefunction. There has been criticismof the fact that mathematicalprobabilityis usually prescribednot only to be additivebut even to be countablyadditive. that is.The question whether real world probabilityis countably additive. into a measurablespace (S'.-Sept. . The space S'.variables. . s) x(t. a positive countablyadditive set function.asks whether a mathematicalmodel of real world probabilisticphenomena necessarilyalwaysinvolves countablyadditiveset functions. of a numberx between 0 and 1. enduringrelics of the historicalbackgroundof probability theory.This extensionwas not developed in order to provide a basis for probability.Some probabilistic slang will be needed. Thus it was 28 years before Lebesgue'stheory was extended far enough to be adequate for the mathematicalbasis of probability. '). A measurewith this normalizationis a probabilitymeasureA randomvariableis a measurablefunctionfrom a probability space (S.however.1] is provided with Lebesgue measure. turned out to be the final essentialresult needed to formulatethe basic mathematicalprobabilitydefinitions. .But there have been very few applicationsof such set functionsin either mathematicalor nonmathematical contexts.(S'. had 1996] THE EVOLUTION OF .) is the randomvariable defined by the value at time t of a function in S. It was well knownwhat the joint distributionsof the randomvariablesof a Brownian motion process should be.But there was as yet no proof of the existence of a stochastic process. a mathematical construct. A correctproof was given a year later by Faber. and 2-n is also the total length (= Lebesgue measure)of the finite set of intervalswhose points x have dyadicrepresentationswith a specifiedsequence of 0's and l's in the first n places.by findingcorrespondingdistributionsfor a certaindiscrete randomwalk and then going to the limit as the walk steps tended to 0. The Brownianmotion stochastic process in R3 is the mathematicalmodel of Brownianmotion.variables which have exactly the distributionsused in calculating coin tossing probabilities. with the desired properties.188.That is. like his fundamentalwork in potential theory. an exampleof a new kind of convergencetheorem in probability. By 1900. Thus a mathematicalversionof the law of large numbersin the coin tossingcontextis the existencein some sense of a limit of the sequenceof function averages{(x1 + *@+xn)/n. Observe that there was no question about the existence of Brownianmotion. and if it was approximable by his randomwalks. in a tossing experiment.Brownianmotion is observableunder a microscope.It omits severalintermediate remarkedtactfully:<<Borel's arguments and assumes certain results without proof. vanishing. Some of the quotationsgiven above indicate that they not only need not but should not. such as that of the maximumchange duringa time interval. Wiener (1923) constructed the desired Brownianmotion process. Observethat (fortunately)pure mathematiciansneed not interpretthis theoremin the real worldof real people tossingreal coins. . and much simpler proofs have been given since. Classical elementaryprobabilitycalculations implythat this sequence of averagesconvergesin measureto 1/2. by applying the Daniell approachto measure theory to obtain a measure with the desired propertieson a space S of continuousfunctions:if x(t.210 on Sat. . and a (normalized)Brownian motionprocessin R is the processof a coordinatefunctionof a normalizedprocess in R3.initially. Bachelier'sresults remainedunnoticedfor years. but a stronger mathematicalversion of the law of large numbers was the fact deduced by Borel-in an unmendablyfaulty proof-that this sequence of averagesconverges to 1/2 for (Lebesguemeasure)almost everyvalue of x. More precisely. the stochasticprocess of these randomvariablesis a stochasticprocess with sample functionsthe membersof S.>>]This theorem was an importantstep. The process is normalizedby supposingit starts at the originof a cartesiancoordinatesystemin R3. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions 591 . Bachelier had even derived various importantdistributionsrelated to the Brownianmotion process in R.what Bachelierderivedwere distributionsvalid for a Brownianmotion process if in fact there was such a thing as a Brownianmotion process. Daniell (1918) used a deep approachto measure theory in which integralsare defined before measuresto get a (ratherclumsy)approachto infinite sequencesof randomvariablesby way of measuresin infinite dimensionalEuclideanspace.A (normalized)Brownian motion process in RN is a process defined by N mutuallyindependentBrownianmotion processes in R. and it had been taken for granted that in a proper mathematicalmodel the class of continuous paths would have probability1. 2-n is the probabilityassignedto the event that. and in fact were rediscovered several times. n2 1}. Wiener'swork. [Frechet proof is excessivelyshort. the motion of a microscopicparticlein a fluid as the particleis hit by the molecules of the fluid. This content downloaded from 213.the first n tosses yield a specifiedsequenceof heads and tails. now sometimes called the Wiener process.233. and with the joint randomvariabledistributionsthose prescribedfor the Brownian motion process. indexed by the finite subsets of X. (b) Random variables on (S.More generally. the measurable space can be a completeseparablemetricspace togetherwith the (r algebra of its Borel sets) and a mutuallycompatibleset of distributionson stn. Steinhaus (1930) demonstrated that classical arguments to derive standard probabilitytheorems could be placed in a rigorous context by taking Lebesgue measureon a linearintervalof length 1 as the basic probabilitymeasure. for fixed B.) onS (c) Conversely.) on S'm is the m-dimensional distribution induced by the n-dimensional distribution of x(t1. with state space S'. with state space S'.P). the space of all functionsfrom > into S'. are the counterpartsof functions of real world observations. and obtained the requiredrandomvariablesas the coordinatefunctionsof S'>. new probabilitiesare obtained.. and a suitablyrestrictedmeasurablespace (S".. conditional probabilities and expectations relative to given values of those randomvariablesare needed.and expectations of randomvariablesfor given B are computedin terms of these new conditionalprobabilities. the scorn of rigorous mathematicsexpressed by some nonmathematicians would be justified.233.No new proofs were required..). given an arbitrarycollection of random variables.. let IFbe the smallest sub (r algebra of S relative to which all the given randornvariables are 592 THE EVOLUTION OF .little immediate influence because it was published in a journal which was not widely distributed.If (S. all that was requiredwas a proper translationof the classicalterminologyinto his context. given an event B of strictlypositive probability. X. P).S. the joint distributionof x(t1.Suppose {x(t.') (for example. x(tm. and expectationsof randomvariablesas their integrals. A set of n of the process randomvariableshas a probabilitydistributionon stn. and if a collection of randomvariablesis given. t E >} defined on it.). If this were all mathematizationof probabilityby measure theory had to offer.To prove this result he constructeda probabilitymeasureon a (r algebraof subsetsof the productspace S'>. The sets in S are the mathematicalcounterpartsof real world events.t e>} is a stochastic process on a probabilityspace (S. . for integers n 21. P) is a probability space. that is of individual(possible) real world observations.It was an aspect of his genius that he carriedout his Brownian motion researchthen and later without knowledgeof the slang and some of the useful elementarymathematicaltechniquesof probabilitytheory. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions .188. (e) The expectationof a numericallyvalued integrablerandomvariable is its integralwith respect to the given probabilitymeasure.interpreting random variables as Lebesgue measurable functions on this interval.)..210 on Sat. . .Kolmogorov(1933) constructedthe following mathematicalbasis for probabilitytheory. X. . functionsof the values assigned to the conditioningrandomvariables. Such finite dimensional distributionsare mutuallycompatiblein the sense that if 1 < m < n. (f) The classicaldefinitionof the conditionalprobabilityof an event (measurable set) A.). x(tn..the points of S are counterpartsof elementaryevents. . there is a probability space and a stochasticprocess {x(t. This content downloaded from 213. P). (a) The context of mathematicalprobabilityis a probabilityspace (S. with the assigned joint random variable distributions. In this way.-Sept. [Aug.is P(A r) B)/ P(B).Kolmogorovproved that given an arbitraryindex set X. 8 Kolmogorov's1933 monograph. the authorrecallshis student experiencein 1932when there were professorial disapprovingremarks on the extreme generality of a seminarlecture givenby Sakson what is now called the Vitali-Hahn-Sakstheorem. a theorem which has since become an importanttool in probabilitytheory.The existence of such a randomvariable. P).to learn that in discussingFourierseries they can be accused of discussingprobabilitiesand expectations.233.he does not simplyrefer to Lebesguebut gives a detailed proof of what he needs. A reasonableinterpretationof a measurable real valued function of the given collection of random variables is a measurablefunctionfrom(S. and similarlywhen he defines the expected value of a randomvariable he does not simplystate that it is the integralof the randomvariablewith respect to the given probabilitymeasure.he must have thoughtthat mathematicianswere not familiarwith measuretheory.The conditionalexpectationof x relativeto a collectionof randomvariablesis defined as the conditionalexpectationof x relativeto the (r algebrageneratedby conditionson the randomvariables. as they are under the added hypothesis that they have a bivariateGaussian distribution.The idea that a (mathematical)randomvariableis simplya function. As confirmationof Kolmogorov'scaution in invokingmeasuretheory.A prominentstatistician in 1935 wonderedwhether two orthogonalreal valued randomvariableswith zero means (integrals)are necessarilyindependent.however.measurable. relativeto a (r algebra(Cof measurablesets. Even as late as 1933.Later in the monograph. This content downloaded from 213.and should be expandedin the view of some probabilists.He was rather surprisedby the exampleof the sine and cosine functionson the interval[O.with probabilitymeasure defined as Lebesgue measure divided by 2qr. is a randomvariablewhich is measurable relativeto (Cand has the same integralas x on everyset in (C. 9 Expansionbackwardsof the Kolmogorovbasis. .210 on Sat. IF)into R.but he actuallydefines the integral.and its uniquenessup to P-nullsets.2qr]. is assuredby the Radon-Nikodymtheorem. Kolmogorov's1933 exposition paints a discouragingpicture of mathematical progress In the first pages of his monographhe states explicitlythat real valued randomvariables are measurablefunctions and expectationsare their integrals. Some analysts may be gratified.188. when he needs Lebesgue'stheorem allowingtaking limits of convergentfunction sequences under the sign of integration. Instead he actuallydefines measurabilityof a real valued function.seemed ratherhumiliatingto some probabilists. These two functions.orthogonaland with zero means but not independent.with no romantic connotation.This S algebrais the (r algebra generated byconditions imposed on thegivenrandomvariables. The Kolmogorovconditional expectationof a real valued integrablerandomvariablex on (S. A conditionalprobabilityof a measurableset A is defined as the conditional expectationof the randomvariablewhich is 1 on A and Oelsewhere. Kolmogorov'sbasis for mathematicalprobabilitycan be expanded.who want to start with some not necessarilynumericalmathematical version of the confidence of observers that certain events will occur. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions 593 . .In fact in the body of his monograph.are not the kind of random variables probabilistswere used to.he does not simplyrefer back to the first pages of the monographand say that a randomvariableis a measurable function. some humiliated. and to 1996] THE EVOLUTION OF . X.] It was some time before Kolmogorov'sbasis was accepted by probabilists.when he comes to the definitionof a real valued randomvariable. [He also recalls that he did not understandthe point of Kolmogorov'smeasure on a functionspace until long after he had read the monograph. the partialsums Y1. xn_1 vanishes (P) almost everywhere.Y2. t E >} is an intervalof the line. 11 Reluctanceto accept measuretheoly by probabilists. and if the state space of the randomvariablesis R.If these randomvariablesare squareintegrable. because all the measure basis to probabilitydoes is to give a formal precise mathematicalframeworkfor the classical calculationsand their present refinements. The followingquotationis an exampleof the reluctanceof some mathematiciansto separate the mathematicsfrom the context that inspiredit.Probabilitytheory suggested new relations between functions.this condition is equivalentto the condition. but need proceed postulationallyto numericalevaluationsof this confidence.. PersonallyI preferas littlefiwssas possiblebecauseI firmly believethatprobabilitytheozyis more closelyrelatedto analysis. In fact. Kac (1959) How muchfiwssovermeasuretheozyis necessazyfor probabilitytheotyis a matterof taste.This condition on a sequence of functions means that in a reasonable sense the sequence of partial sums of the given sequence is the counterpartof a fair game.P) and suppose that the conditionalexpectation of xn given x1. that is.much strongerthan mutualorthogonality. but any approachto the subjectwhich ends with a justificationof the classicalcalculations and is mathematicallyusable.To understandthe difficulty. for example. Bernstein (1927) seems to have been the first to treat such sequences systematically.the probabilistic context served only to suggest mathematicsand was not an integral part of the mathematics.and if Z is a subset not be measurableif Z is uncountable. whatever the choice of joint distributionsof the process randomvariables. and finallyto additivity. A clumsy approachwas proposed by Doob (1937) but a more usable one was not devised until after 1950. x2. will end with Kolmogorov'sbasis. xn. [Aug. For example consider the sequence x1. the integral of xn over any set determinedby conditions on the preceding random variablesvanishes. 12 New relations between functions made possible by the mathematizationof probability. . are characterizedby the propertythat the expectationof Yn 594 THE EVOLUTION OF .. There was considerable resistance to the acceptance and exploitationof measure theory by probabilists. . for example to potential theory and partial differentialequations.. of real valued integrablerandomvariables on a probabilityspace (S.If boundednessand continuityof sample functionsare to be discussed.of >. . .210 on Sat.188. 10 Uncountableindex sets. the function s supt>x(t.-).Although such applicationswere made in the past before the acceptance of measure theory as the basis of probability.for n > 1.The meaningof solutionsas probabilitiesand expectationscould not be formulatedand exploited. both in Kolmogorov'sday and later. Si. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions . . .Such an analysismay be enlightenedin discussingthe appropriateness of mathematicalprobability as a model for real world phenomena. howeverphrased. .that xn is orthogonal to every square integrable function of x1. ...some modificationof the probabilityrelationsof the randomvariablesof a stochasticprocess should be devised to make such suprema measurablefunctions. . s) is measurableif Z is countable. the class of continuoussample functions may not be measurable.observe that if the index set > of a stochasticprocesswith state space 1Ris an interval.This difficulty arises in the processes derivedby the Kolmogorovconstructionof a measureon a function space.233.-Sept. If the index-set > of a stochastic process {x(t. This frameworkhad made it possible to applymathematicalprobabilityin many other mathematicalfields. . . physicsand statisticsthan to measuretheotyas such. This content downloaded from 213. These authorsare in the interestingsituation stochastic processes they call it analysis if the family of functions x(t.. ( 13.233. processes in 1944withstochastic integral Protter By developinghis techwithpurelyprobabilistic diffusions wasableto studymultidimensional overtheanalyticmethodsof Feller. (t. immediate the on depends only very proved has others) by honor in his (named 1906. If x is an orthogonalsequence of functions. s) as s varies is studied.1) The orthogonal series converges almost everywhere if either (Mensov-Rademacher)(13. relative to xn_1. 1996] THE EVOLUTION OF .1104 101 WestWindsor Illinois61801-6663 Urbana.and if so whether measuretheoryshould also be evicted.Apt.This point of view is expressedin the followingquotation. . These Markov the with of sequences random variables is the class probabilithe conditional 2 1 n when that the fact sequencesare characterizedby for xn those to everywhere almost are equal ties for xn relative to xl. the given present. s) as in considering relative to Y1 XYn-lis equal to Yn-l almost everywhereon the probability first used explicitlyby Ville space.1) is strengthenedto n < + oo.for example.generalizingand includingclassicalpotential theory. Markov The past. in Markov special case by fruitful.an improvement The following remark on the convergence of a sum of orthogonal functions illustratesthe difficultyin separating(mathematical)probabilityfrom the rest of analysis.The measure space is a probabilityspace. E (rn2 log2 Road.on a probabilitymeasure space. niques.called martingales. have had many applications. Another property. math.2) or (Levy. but call it probabilityand definitelynot analysisif the familyof functions x(. 13 What is the place of probabilitytheoly in measuretheoly. Roughly very in a introduced property. The reader shouldjudge which of these results is measuretheoretic and which is probabilistic. they regard discussions of distributionsand associatedquestionsas analysis. 1937) the condition (13. . .for example class of sequences of important to derivation. xn_1 past.then (Riesz-Fischer)Exn convergesin the mean if E 2 < + (13. s) x(t.but that if one deals with samplesequencesand samplefunctionsthen the subjectis probabilitybut not analysis.uiuc. and if xn2has integral (rn2. partial to (1939).210 on Sat. Processeswith this property. but with trivial changes the discussionis valid for any finite measurespace.and to potential theory.whether there is any point in evicting mathematicalprobability from analysis. This content downloaded from 213.but not discussionsin termsof sample functions. differentialequations. leading in the second half of the centuryto a probabilistic potential theory. Ito as integrands. . of the the influence speaking.edu [email protected]) is kept but the orthogonalitycondition is strengthenedto the conditionin Section 12. . More precisely. 07 Nov 2015 11:14:07 UTC All use subject to JSTOR Terms and Conditions s9s .) as t varies is studied.188. and more generally in analysis? It is considered by some mathematiciansthat if one deals with analytic properties of probabilitiesand expectationsthen the subject is part of analysis.that in consideringa functionof two variables. Documents Similar To Doob, The Development of Rigor in Mathematical Probability (1900-1950).pdfSkip carouselcarousel previouscarousel nextili10ExtremeEventDependenceBibliography 5-18-06Ch2Stein10.1.1.70.6249cantor.pdfSyllabus B.Tech-ECE-2016chan vivaMsg 00013Math 3FM3 Fall 2012 Course OutlineSummarySimulation in FinanceDerivatives13820-15995-1-PBlec7.pdfProbability 3Itoˆ versus Stratonovich calculus in random population growthChapter 1 Integrals and the Haar Measure1. AOPcontinuous markov chainnspch6Course Descriptions-Required CoursesStochastic Simulation in Agricultural EconomicsMatLASR_2003integrablity help 3.pdf6 Annotated Ch5 Discrete Random Variables IDennis Essers - A Review of Brian Arthur s Increasing Returns and Path Dependence in the EconomyComparison of 3 Algorithms of Levy GenerationFinancial Engineering Part 2 Lecture 1 2012 2w12stat302assign3More From FardadPouranSkip carouselcarousel previouscarousel nextDaoist Perspectives on Chinese and Global Environmental ManagementOmar Khayyam and the cubic equation.pdfJohn B. Walsh, Notes on Elementary Martingale TheoryFred Richman - Constructive Mathematics Without Countable Choiceپروژه "شبیه سازی"پروژه جبرخطی - زنجیره مارکوفاصل ناورداییTips for Success in Undergraduate Math CoursesDifferntial Forms - Terence TaoImo 2010 (Korean)شهود چيست؟ استدلال چيست؟Footer MenuBack To TopAboutAbout ScribdPressOur blogJoin our team!Contact UsJoin todayInvite FriendsGiftsLegalTermsPrivacyCopyrightSupportHelp / FAQAccessibilityPurchase helpAdChoicesPublishersSocial MediaCopyright © 2018 Scribd Inc. .Browse Books.Site Directory.Site Language: English中文EspañolالعربيةPortuguês日本語DeutschFrançaisTurkceРусский языкTiếng việtJęzyk polskiBahasa indonesiaMaster your semester with Scribd & The New York TimesSpecial offer for students: Only $4.99/month.Master your semester with Scribd & The New York TimesRead Free for 30 DaysCancel anytime.Read Free for 30 DaysYou're Reading a Free PreviewDownloadClose DialogAre you sure?This action might not be possible to undo. Are you sure you want to continue?CANCELOK