Currently published HOL formalizations of measure theory concentrate on the Lebesgue integral and they are restricted to realvalued measures. We lift this restriction by introducing the extended real numbers. We define the Borel σ-algebra for an arbitrary type forming a topological space. Then, we introduce measure spaces with extended real numbers as measure values. After defining the Lebesgue...

The function u(t) need not be a member of C, but can be any Borel measurable function defined on OSj/^1, and may even be permitted to take on infinite values. The Wiener integral of a functional is simply the Lebesgue integral of the functional with respect to Wiener's measure [4]1 in C. This measure is not invariant under translations, but is in other respects a Lebesgue measure based on inter...

We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbol...

It is shown that Schrödinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points of the interval.

the euclidean group (rn,+) where (n?n, plays a key role in harmonic analysis. if we consider the lebesgue measure ()nd?xr as the haar measure of this group then 12(2)()nd?x=d?rr. in this article we look for lca groups k, whose haar measures have a similar property. in fact we will show that for some lca groups k with the haar measure k?, there exists a constant such that 0kc>()(2)kkk?a=c?a for ...

Let $K$ be a locally compact hypergroup with left Haar measure and let $L^1(K)$ be the complex Lebesgue space associated with it. Let $L^infty(K)$ be the dual of $L^1(K)$. The purpose of this paper is to present some necessary and sufficient conditions for $L^infty(K)^*$ to have a topologically left invariant mean. Some characterizations of amenable hypergroups are given.

We consider dynamical systems on compact manifolds, which are local dif-feomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbo...

§1 Preliminaries: Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §2 Riemann Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 §3 Lebesgue measure zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 §4 Definition and Properties of the Lebesgue Integral . . . . . . . . . . 7 §5 The spaces L(R) and L(R) ....