In 1918 and 1919 Daniell's papers on what is now called the Daniell integral appeared. As an application he constructed the most general finite measure of Borel sets in Euclidean space of countable dimensionality. In 1933 Kolmogorov, in his formalization of probability as measure theory, rediscovered the Daniell result in constructing measures on Euclidean space of arbitrary dimensionality. Aft...

In this tutorial, I will introduce probability theory from the perspective of lattice theory where the fundamental notion is based on a partial ordering of logical statements ordered by implication. This construction forms a Boolean lattice, which is related to the fact that the corresponding logical operations form a Boolean algebra. I generalize the concept of logical implication to degrees o...

The renormalization group has played an important role in the physics of the second half of the twentieth century both as a conceptual and a calculational tool. In particular it provided the key ideas for the construction of a qualitative and quantitative theory of the critical point in phase transitions and started a new era in statistical mechanics. Probability theory lies at the foundation o...

This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. David Freedman and Philip Stark’s concept of model based probabilities is examined and is used as a bridge between the formal theory and applications.

A sequence of elements an of R d converges to a limit a if and only if, for each ǫ > 0, the sequence an eventually lies within a ball of radius ǫ centered at a. It’s okay if the first few (or few million) terms lie outside that ball— and the number of terms that do lie outside the ball may depend on how big ǫ is (if ǫ is small enough it typically will take millions of terms before the remaining...

Physicists have proposed a wealth of new ideas concerning the collective behavior of large families of random variables, in which they have discovered the emergence of new types of order. The quest for mathematically rigorous proofs of their conjectures is likely to challenge and stimulate probabilists for years to come.

Define f (k, l) as the least integer so that every graph having f(k, 1) vertices contains either a complete graph of order k or a set of l independent vertices (a complete graph of order k is a graph of k vertices every two of which are connected by an edge, a set of I vertices is called independent if no two are connected by an edge) . Throughout this paper c1, c2, . . . will denote positive a...

We compare two approaches to probability on systems of fuzzy sets—measures on tribes as introduced by Butnariu and Klement [3] and measures on MV-algebras with product studied by Riečan and Mundici [21]. We show that these two approaches overlap significantly and they confirm the original proposal of probability measure on fuzzy sets by Zadeh [24].