Approximately continuous functions were first introduced in connection with problems of differentiation and integration. The main idea of the approximate continuity of a function f is that the continuity conditions are true not necessarily everywhere but only almost everywhere with respect to some measure, e.g., Borel measure or Lebesgue measure. At the same time, it is known that functions tha...

BACKGROUND Continuity of care is widely acknowledged as important for patients with multi-morbidity but simple, service-orientated indices cannot capture the full impact of continuity in complex care delivery systems. The patient's perspective is important to assess outcomes fully and this is challenging because generic measures of patient-perceived continuity are lacking. We investigate the Ch...

We present a simple and clear foundation for finite inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying lattices of logical statements in a way that satisfies general lattice symmetries. With other applications such as measure theory in mind, our derivations assume minimal symmetries, relying on neither negation nor continui...

as was introduced by Bollobás, Janson and Riordan in [1]. Here S is a separable metric space and μ is a Borel probability measure on S . No relationship is assumed between x (n) i and x (n′) i . To simplify notation we shall from now on write (x1, . . . , xn) = (x (n) 1 , . . . , x (n) n ). We begin by recalling some basic definitions and assumptions from [1]. For each n let (x1, . . . , xn) be...

Let T be an ergodic automorphism of the d-dimensional torus T, and f be a continuous function from T to R. On the probability space T equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f ◦T )i≥1 under some condition on the modulus of continuity of f . The proofs are based on new limit theorems and new inequalities for non-...

We study perturbations of a stochastic program with a probabilistic constraint and r-concave original probability distribution. First we improve our earlier results substantially and provide conditions implying Hölder continuity properties of the solution sets w.r.t. the Kolmogorov distance of probability distributions. Secondly, we derive an upper Lipschitz continuity property for solution set...