Stochastic processes with an integral-valued parameter

The purpose of this paper is to set up the measure relations of the most general stochastic process and to discuss the properties of the conditional probability functions of the processes depending on a parameter running through integral values.In particular, the study of temporally homogeneous processes of this type is shown to be essentially the study of measure preserving transformations.The well known results in the latter field are applied to develop and extend the theory of Markoff processes from a new point of view.Throughout the paper, any non-negative completely additive function of point sets, denned on a Borel field of setsf of some abstract space ß will be called a probability measure if the space ß is itself in the field of definition and if the set function is defined as 1 on ß.1. Probability measures defined on spaces of infinitely many dimensions.Let fio (X) be any abstract space, consisting of elements w0 (x), and let Fo,0 (Fx) be a Borel field of subsets of ß0(X).We shall suppose that ß0 e Fa", and X e Fx.If/(w0) is a function defined on ß0 which takes on values in X, and if the ß0-set defined by the condition /(w0) e E is in FUo for every set E of Fx, then the function/(w0) will be called measurable on ß0.Let {/"(wo)} be any sequence of such measurable functions, where the subscript « ranges through any aggregate zA, not necessarily denumerable.If a probability measurePo(A0) is defined on the sets A0 of FUo, the measurable function/n(w0), considered from the standpoint of probability, is a chance variable xn.The following method of analyzing the mutual relations of such a set of chance variables has been used, more or less explicitly, in recent years.J Consider the space ß whose points are the aggregates oj: {xn}, « e zA, xn e X. § If n runs through all real numbers /, ß consists of all functions of the real variable t, * Presented to the Society, April 9, 1937; received by the editors July 26, 1937.f A field of sets is a collection of sets E containing, with E, and Et, their sum E,-\-E% and difference E\-E, ■ Ei.A Borel field of sets is a field which contains with Ex, Ei, • • • their sumj^, E¡. % Cf.Doob (I); Hopf (I); Khintchine (I); Kolmogoroff (II, pp.24-30); Levy (I and numerous papers) ; Lomnicki and Ulam (I) ; Paley and Wiener (I, chaps.9 and 10, and earlier papers by Wiener).The Roman numerals refer to the bibliography at the end of the paper.§ Each subscript n determines a coordinate x", and the space Ü is thus a Cartesian space with a dimension corresponding to each element of <¡A.87 * The field F was defined at the beginning of the proof.t Cf.H. Hahn, Annali délie Università Toscane, Pisa, (2), vol. 2 (1933), pp.433-436.t Cf. §1.* Cf., however, the note on p. 88 in accordance with which it may sometimes be necessary to define a stochastic process using a subspace of fi rather than fi itself, as in Doob (II).f The extension theorem used in the proof of Theorem 1.1 defines P(A) in precisely this way.Î The complement of any set A will be denoted by CA throughout this paper.§ The corresponding theorems for Borel and Lebesgue measurable functions are discussed by de la Vallée Poussin in his book Intégrales de Lebesgue, Fonctions d'Ensemble, Classes de Baire, 2d edition, Paris, 1934, pp.34-40.