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Measure Theory
The concept of a measure is a generalization and formalization of length, area, volume, and other common notions such as mass and probability of...
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Abstract Measure Theory
The objective of this chapter is to extend measure theory by generalizing the concept of measure to include more general sets other than sets in...
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Zeno Product Formulæ
This chapter is devoted to study a link between Trotter, or Trotter-Kato product formulæ, and product formulæ including projections. Formally, the...
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The Haar Measure
It is well known that every locally compact group admits a (one sided) translation invariant Haar measure. Applications of the Haar measure in...
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Product Space and Coupling
Coupling is a method for combining two random variables defined on separate probability spaces in order to study them simultaneously. It is the...
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Measure Theory
There is a motivation to generalize the notion of length to another concept that could capture length in a more general setting. In principle, the...
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Unitary Product Formulæ
In this chapter, we first recollect the main results about convergence of the unitary Trotter product formula in the strong operator topology, Sect....
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Multiplicative arithmetic functions and the generalized Ewens measure
Random integers, sampled uniformly from [1, x ], share similarities with random permutations, sampled uniformly from S n . These similarities include...
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The Hartley–Bessel function: product formula and convolution structure
This paper explores a one-parameter extension of the Hartley kernel expressed as a real combination of two Bessel functions, termed the...
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Prelude: Viète’s Product
We prove Viète’s infinite product for \(\uppi \) using some...
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Segmentation in Measure Spaces
We consider an abstract concept of perimeter measure space as a very general framework in which one can properly consider two of the most...
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General Measure and Integration
The first five sections of this chapter are dedicated to the study of measure theory, measurable functions on a measure space, general integration...
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Strong Proximality for Discontinuous Skew-Product Actions of Amenable Semigroups
The invariant Borel probability measures play a crucial role in the study of the notion of the support of topological dynamical systems....
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Itô-Wentzell-Lions Formula for Measure Dependent Random Fields under Full and Conditional Measure Flows
We present several Itô-Wentzell formulae on Wiener spaces for real-valued functional random field of Itô type that depend on measure flows. We...
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Topological aspects of the space of metric measure spaces
Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces...
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Product formulas and convolutions for Laplace–Beltrami operators on product spaces: beyond the trivial case
We introduce the notion of a family of convolution operators associated with a given elliptic partial differential operator. Such a convolution...
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On the Extension of a Family of Projections to a Positive Operator-Valued Measure
AbstractIn this paper, the problem of constructing a measure which accepts values within a positive cone of bounded operators in a Hilbert space is...