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The essential properties of general Lebesgue outer measure are discussed. The complete measure space, consisting of the general Lebesgue outer measure restricted to the measurable sets, is developed and this measure is shown to be unique. Two characterizations of measurable sets are discussed. The Borel sets are investigated in the plane and more generally, in n-space, and it is shown that the sigma-algebra of Borel sets is equal to the product sigma-algebra of Borel sets on the line. Finally, the interrelationships between Lebesgue measure in the plane and the product measure of Lebesgue measures on the line are investigated. It is shown that the sigma-algebra of Lebesgue measurable sets properly contains the product sigma-algebra and that these two measures agree on the product sigma-algebra. It is also proven that the sigma-algebra of Lebesgue measurable sets is the completion of the product sigma-algebra. Examples are provided to illustrate that the product measure spaces discussed are not complete as well as an example of a subset of the plane which is not Lebesgue measurable. (Author)

In Kelley, Reed, and Root (1960), it was shown that if the log of the likelihood ratio for detecting a signal in stationary Gaussian noise is phi sub T when the data is z(t), t epsilon (-T,T) and phi when the data is z(t), t epsilon (minus infinity, infinity), then Var phi sub T increases monotonically to Var(phi) as T approaches infinity. Recently, this result was extended to vector-valued stationary noise processes by Salehi (1968). In each case the purpose was to show that the probability of error for t epsilon (-T,T) converges to the probability of error for t epsilon (minus infinity, infinity). The purpose of this paper is to show that both of the results above are but special cases of a more fundamental result to be given here. (Author)

Let f(x) be a real or vector valued random process whose parameter set R is a bounded set in Euclidean n-space. Following the usual terminology, a version of f(x) is any process f tilde (x) defined on R which satisfies P(f(x) = f tilde (x)) =1 for each x epsilon R. It is sometimes useful to assert that there exists a version of f(x) which is (with probability one) continuous, Holder continuous, or perhaps differentiable in some component. Also estimates of the type P(sup (x epsilon R)/f(x)/> epsilon) may be desired for these versions. Illustrative examples are given., Sponsored in part by National Aeronautics and Space Administration, Washington, D. C. Grant 40-002-015.

Let 1 denote the vector space of all sequences of real numbers with the topology of coordinate-wise convergence. For 0 < p < infinity l sub p denote the subset of l consisting of all sequences x sub i which have the summation, from one to infinity of the (absolute value of x sub i) to the p power finite. The main efforts in the paper are to generalize Bochner's theorem and the Levy-continuity theorem to these l sub p spaces. (Author)

A generalization of the Strong Law of Large Numbers to weakly orthogonal sequences of Banach-Space valued random variables is given. (Author)

A function psi(t) is called denumerably lower continuous (dlc) if it is continuous on a dense denumerable set t sub 1, t sub 2,... and if for all t psi(t) = lim as t sub alpha approaches t of (inf psi(t sub alpha)). It is shown that every process which is almost certainly connected can be realized in the space of dlc functions. (Author)