Regularity properties of certain families of chance variables

Abstract

Let / vary in any simply ordered set, and let {xt} be a family of chance variables.We shall say that the chance variables have the property £ if whenever *!<••• </"+],with probability 1.The subscript n here is any positive integer.It will always be supposed that the given chance variables have expectations, so that the conditional expectations involved in the definition of the property £ will always exist.We shall first make a few introductory remarks on the general family of chance variables with the property £.Then in §1 it will be supposed that t takes on integral values, and the convergence properties of such sequences of chance variables will be discussed in detail.In §3 it will be supposed that / runs through the real numbers.Before discussing this case, it is necessary to investigate the justification for the use of such descriptive terms as continuity, boundedness, and so on, as applied to the random function xt.This investigation is made in §2, in the general case, without the hypothesis of the property £, and it is found that the above terms can always be made meaningful, and given their usual meanings, at the cost, however, in some cases, of introducing infinite-valued functions.In the following, we shall always suppose that the xt are measurable functions defined on a space Q, on certain sets of which a measure function is defined.That this can always be done, and how this is to be done, was shown by Kolmogoroff [5,.The space fl, following Kolmogoroff, will be taken to be the space of real-valued functions of t.Integration in terms of probability measure will be denoted by f ■ ■ ■ dP, and integration will be over all space, unless the domain of integration is otherwise specified.The quali-* Presented to the Society, December 28, 1939, under the title On a certain type of family of chance variables; received by the editors December 11,1939.t We shall use the notation E[y] for the expectation of the chance variable y, and E [yi, ■ ■ -,yn;y] for the conditional expectation of y for given yi, ■ ■ ■ , yn, a function of yi, • ■ ■ , yn.If the y, are not finite in number, the notation will be modified accordingly.We shall assume the definitions of Kolmogoroff [5, for these conditional expectations.(References in brackets are to the bibliography at the end of the paper.