It is shown that paradoxes arise in conditional probability calculations, due to incomplete specification of the problem at hand. This is illustrated with the Borel and the Kac-Slepian type paradoxes. These are significant in applications including Bayesian inference. Also Rényi's axiomatic...

The classical Kolmogorov theorem on the existence of stochastic process has been generalized in several directions following its abstract formulation by Bochner. In the first half of the paper a unified exposition of the key results of the existing work is given. The second half consists of some...

If ([Omega], [Sigma], [mu]) is a general measure space, where [Sigma] is a [delta]-ring, let Lp([Sigma]) be the corresponding Lebesgue space. One of the authors has defined and studied the properties of the (generalized) conditional expectation E[Lambda] on Lp([Sigma]) relative to a [delta]-ring...

A detailed study of the structure of conditional expectations and conditional probability measures is presented. Some characterizations of conditional expectations as a subclass of projection operators on Banach function spaces, and similarly conditional probabilities as a subclass of vector...

In the first part of this paper, nonlinear prediction theory of vector valued random variables in Orlicz spaces is presented. The spaces need not be reflexive and the results of this part are essentially best possible for these spaces. The second part considers operator valued martingales in the...