By Henry M. Paynter

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However, Radon random variables form a somewhat too restrictive setting for other types of questions. For example, if we are given a sequence (Xn) of real valued random variables such that supn IXnl < 00 almost surely, and if we ask (for example) for the integrability properties or the tail behavior of this 44 2. Generalities on Banach Space Valued Random Variables supremum, we are clearly faced with a random element of infinite dimension but we need not (and in general do not) have a Radon random vector.

Of course, these considerations raise some nontrivial measurability questions as soon as T is no longer countable. A priori, a random process X = (XthET is almost surely bounded or continuous, or has almost all its trajectories or sample paths bounded or continuous, if, for almost all w, the path t - Xt(w) is bounded or continuous. However, in order to prove that a random process is almost surely bounded or continuous, and to deal with it, it is preferable and convenient to know that the sets involved in these definitions are properly measurable.

Moreover, every dense sequence S in T can be chosen as a separable set. The preceding hypotheses will always be satisfied when we will need such a result so that we freely use it below. Summarizing, the study of the almost sure boundedness and continuity of random processes can essentially be reduced with the tools of essential supremum or separable version to the setting of a countable index set for which no measurability question occurs. In our first part, we will therefore basically study integrability properties and tail behaviors of supremum of bounded processes indexed by a countable set.

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