A multivalued version of the Radon-Nikodym theorem, via the single-valued Gould integral

Some topics concerning the Gould integral are presented here: new results of integrability on finite measure spaces with values in an M-space are given, together with a Radon-Nikodym theorem relative to a Gould-type integral of real functions with respect to a multisubmeasure.Comment: 19 page

Bayesian model-based spatiotemporal survey design for log-Gaussian Cox process

In geostatistics, the design for data collection is central for accurate prediction and parameter inference. One important class of geostatistical models is log-Gaussian Cox process (LGCP) which is used extensively, for example, in ecology. However, there are no formal analyses on optimal designs for LGCP models. In this work, we develop a novel model-based experimental design for LGCP modeling of spatiotemporal point process data. We propose a new spatially balanced rejection sampling design which directs sampling to spatiotemporal locations that are a priori expected to provide most information. We compare the rejection sampling design to traditional balanced and uniform random designs using the average predictive variance loss function and the Kullback-Leibler divergence between prior and posterior for the LGCP intensity function. Our results show that the rejection sampling method outperforms the corresponding balanced and uniform random sampling designs for LGCP whereas the latter work better for models with Gaussian models. We perform a case study applying our new sampling design to plan a survey for species distribution modeling on larval areas of two commercially important fish stocks on Finnish coastal areas. The case study results show that rejection sampling designs give considerable benefit compared to traditional designs. Results show also that best performing designs may vary considerably between target species

Signed topological measures on locally compact spaces

In this paper we define and study signed deficient topological measures and signed topological measures (which generalize signed measures) on locally compact spaces. We prove that a signed deficient topological measure is $\tau$-smooth on open sets and $\tau$-smooth on compact sets. We show that the family of signed measures that are differences of two Radon measures is properly contained in the family of signed topological measures, which in turn is properly contained in the family of signed deficient topological measures. Extending known results for compact spaces, we prove that a signed topological measure is the difference of its positive and negative variations if at least one variation is finite; we also show that the total variation is the sum of the positive and negative variations. If the space is locally compact, connected, locally connected, and has the Alexandroff one-point compactification of genus 0, a signed topological measure of finite norm can be represented as a difference of two topological measures.Comment: 23 page

Categorifying measure theory: a roadmap

A program for categorifying measure theory is outlined.Comment: AMS-LaTeX + xy-pic. 90 page

Supermartingale Deomposition with General Index Set

We prove results on the existence of Dol\'{e}ans-Dade measures and of the Doob-Meyer decomposition for supermartingales indexed by a general index se

Model interpretation through lower-dimensional posterior summarization

Nonparametric regression models have recently surged in their power and popularity, accompanying the trend of increasing dataset size and complexity. While these models have proven their predictive ability in empirical settings, they are often difficult to interpret and do not address the underlying inferential goals of the analyst or decision maker. In this paper, we propose a modular two-stage approach for creating parsimonious, interpretable summaries of complex models which allow freedom in the choice of modeling technique and the inferential target. In the first stage a flexible model is fit which is believed to be as accurate as possible. In the second stage, lower-dimensional summaries are constructed by projecting draws from the distribution onto simpler structures. These summaries naturally come with valid Bayesian uncertainty estimates. Further, since we use the data only once to move from prior to posterior, these uncertainty estimates remain valid across multiple summaries and after iteratively refining a summary. We apply our method and demonstrate its strengths across a range of simulated and real datasets. Code to reproduce the examples shown is avaiable at github.com/spencerwoody/ghostComment: 40 pages, 16 figure

Finitely additive measures and complementability of Lipschitz-free spaces

We prove in particular that the Lipschitz-free space over a finitely-dimensional normed space is complemented in its bidual. For Euclidean spaces the norm of the respective projection is $1$. As a tool to obtain the main result we establish several facts on the structure of finitely additive measures on finitely-dimensional spaces.Comment: 24 pages; we corrected some misprints, reorganized a bit the introduction and added few comments (in particular one on the Leray projection

Absolutely Summing Operators on non commutative C∗C^*-algebras and applications

Let $E$ be a Banach space that does not contain any copy of $\ell^1$ and \A be a non commutative $C^*$-algebra. We prove that every absolutely summing operator from \A into $E^*$ is compact, thus answering a question of Pe\l czynski. As application, we show that if $G$ is a compact metrizable abelian group and $\Lambda$ is a Riesz subset of its dual then every countably additive \A^*-valued measure with bounded variation and whose Fourier transform is supported by $\Lambda$ has relatively compact range. Extensions of the same result to symmetric spaces of measurable operators are also presented

On reproducing kernels, and analysis of measures

Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive definite kernel is also the covariance kernel of a Gaussian process. Given a fixed sigma-finite measure $\mu$, we consider positive definite kernels defined on the subset of the sigma algebra having finite $\mu$ measure. We show that then the corresponding Hilbert factorizations consist of signed measures, finitely additive, but not automatically sigma-additive. We give a necessary and sufficient condition for when the measures in the RKHS, and the Hilbert factorizations, are sigma-additive. Our emphasis is the case when $\mu$ is assumed non-atomic. By contrast, when $\mu$ is known to be atomic, our setting is shown to generalize that of Shannon-interpolation. Our RKHS-approach further leads to new insight into the associated Gaussian processes, their It\^{o} calculus and diffusion. Examples include fractional Brownian motion, and time-change processes

On subadditive functions upper bounded on a 'large' set

The notion of a shift-compact set in an abelian topological group $X$ plays a significant role in functional equations and inequalities, especially so since each Borel set that is not Haar-meagre, alternatively not Haar-null, is necessarily shift-compact for $X$ completely metrizable (see \cite{BJ} and \cite{BinO8}). Although in general boundedness of a subadditive function does not imply its continuity, here we prove that each subadditive function $f:X\rightarrow \mathbb{R}$ (i.e. with the function satisfying $f(x+y)\leq f(x)+f(y)$ for $x,y\in X$) bounded above on a~shift-compact (non-Haar-null, non-Haar-meagre) set is locally bounded at each point of the domain. Our results refer to \cite[Chapter~XVI]{Kuczma} and papers by N.H.~Bingham and A.J.~Ostaszewski \cite{BO,BinO1,BinO2,BinO6,BinO7}