Related papers: Proofs for Folklore Theorems on the Radon-Nikodym …
In this paper we present some results concerning Gould integrability of vector functions with respect to a monotone measure on finitely purely atomic measure spaces. As an application a Radon-Nikodym theorem in this setting is obtained.
In this paper we present a generalization of the Radon-Nikodym theorem proved by Pedersen and Takesaki. Given a normal, semifinite and faithful (n.s.f.) weight $\phi$ on a von Neumann algebra M and a strictly positive operator $\delta$,…
We show that the stochastic independence of real-valued random variables is equivalent to the conditional uncorrelation, where the conditioning takes place over the Cartesian products of intervals. Next, we express the mutual independence…
We interpret the probabilistic notion of unimodularity for measures on the space of rooted locally finite connected graphs in terms of the theory of measured equivalence relations. It turns out that the right framework for this consists in…
For an image pixel information can be converted to the moments of some basis $Q_k$, e.g. Fourier-Mellin, Zernike, monomials, etc. Given sufficient number of moments pixel information can be completely recovered, for insufficient number of…
The Radon-Nikodym formalism is used to study the structure of the set of positive maps from $\mathcal{B}(\mathcal{H})$ into itself, where $\mathcal{H}$ is a finite dimensional Hilbert space. In particular, this formalism was employed to…
Given a completely positive (CP) map $T$, there is a theorem of the Radon-Nikodym type [W.B. Arveson, Acta Math. {\bf 123}, 141 (1969); V.P. Belavkin and P. Staszewski, Rep. Math. Phys. {\bf 24}, 49 (1986)] that completely characterizes all…
We study the Radon-Nikodym problem for approximately proper equivalence relations and more specifically the uniqueness of certain Gibbs states. One of our tools is a variant of the dimension group introduced in the study of AF algebras. As…
The famous Prohorov theorem for Radon probability measures is generalized in terms of usco mappings. In the case of completely metrizable spaces this is achieved by applying a classical Michael result on the existence of usco selections for…
Using the model of words, we give bijective proofs of Gould-Mohanty's and Raney-Mohanty's identities, which are respectively multivariable generalizations of Gould's identity $$\sum_{k=0}^{n}{x-kz\choose k}{y+kz\choose n-k}=…
The purpose of this article is to formulate a number of probabilistic hidden-variable theorems, to provide proofs in some cases, and counterexamples to some conjectured relationships. The first theorem is the fundamental one. It asserts the…
In this article, we introduce local completely positive $k$-linear maps between locally $C^{\ast}$-algebras and obtain Stinespring type representation by adopting the notion of "invariance" defined by J. Heo for $k$-linear maps between…
The goal of this short note is to provide simple proofs for the "folklore facts" on the sample complexity of learning a discrete probability distribution over a known domain of size $k$ to various distances $\varepsilon$, with error…
Kanade explored the construction of modular companions to $q$-series identities, using the asymptotics of Nahm sums, and Mizuno [Ramanujan J.\ {\bf 66} (2025), Paper No.\ 62, 31] recently obtained a generalization of Kanade's asymptotic…
In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as…
The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability…
An alternative treatment is proposed for the calculations carried out within the frame of Nikiforov-Uvarov method, which removes a drawback in the original theory and by pass some difficulties in solving the Schrodinger equation. The…
In this note we give a new proof of a version of the Besicovitch covering theorem, given in \cite{EG1992}, \cite{Bogachev2007} and extended in \cite{Federer1969}, for locally finite Borel measures on finite dimensional complete Riemannian…
In this paper we discuss generalized Riesz products bringing into consideration $H^p$ theory, the notion of Mahler measure, and the zeros of polynomials appearing in the generalized Riesz product. Formula for Radon-Nikodym derivative…
In this paper, we formulate and prove several variants of the Erd\H{o}s-Tur\'{a}n additive bases conjecture.