Varadhan's integration theorem, one of the corner stones of large-deviation theory, is generalized to the context of capacities. The theorem appears valid for any integral that obeys four linearity properties. We introduce a collection of integrals that have these properties. Of one of them, known as the Choquet integral, some continuity properties are established as well.
Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.
We study properties of variational measures associated with certain conditionally convergent integrals in . In particular we give a full descriptive characterization of these integrals.
Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a -adic path system that defines a differentiation basis which does not possess Ward property.
We study the relationship between derivates and variational measures of additive functions defined on families of figures or bounded sets of finite perimeter. Our results, valid in all dimensions, include a generalization of Ward’s theorem, a necessary and sufficient condition for derivability, and full descriptive definitions of certain conditionally convergent integrals.
We relate the Grothendieck inequality to the theory of vector measures and show that the integral of an inner product with respect to a bimeasure can be computed in an iterative way. We then show an application to the theory of bounded linear operators.
A new approach to differentiation on a time scale is presented. We give a suitable generalization of the Vitali Lemma and apply it to prove that every increasing function f: → ℝ has a right derivative f₊’(x) for -almost all x ∈ . Moreover, .
We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals from one...