1 Product measures Let (X,A , μ) be a σ-finite measure space. Then with A ⊗ A the product σalgebra and μ ⊗ μ the product measure on A ⊗A , (X ×X,A ⊗A , μ⊗ μ) is itself a σ-finite measure space. Write Fx(y) = F (x, y) and F (x) = F (x, y). For any measurable space (X ′,A ′), it is a fact that if F : X×X → X ′ is measurable then Fx is measurable for each x ∈ X and F y is measurable for each y ∈ X...

For a transformation F on a measure space (X,μ), we show that the Perron-Frobenius operator of F can be written by a representation (L2(X,μ), π) of the Cuntz-Krieger algebra OA associated with F when F satisfies some assumption. Especially, when OA is the Cuntz algebra ON and (L2(X,μ), π) in the above is some irreducible representation of ON , then there is an F -invariant measure on X which is...

We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f -invariant set, except in some singular set of neutral points. We prove that if there exists some f -invariant set of hyperbolic points with positive unstable-Lebesgue measure such that for every point in this set the stable and unstable leaves are “long enoug...

Ahlswede, Khachatrian, Mauduit and Sárközy [1] introduced the f -complexity measure (“f ” for family) in order to study pseudorandom properties of large families of binary sequences. So far several families have been studied by this measure. In the present paper I considerably improve on my earlier result in [7], where the f -complexity measure of a family based on the Legendre symbol and polyn...

Let f : {0, 1} → {0, 1} be a Boolean function. The certificate complexity C(f) is a complexity measure that is quadratically tight for the zero-error randomized query complexity R0(f): C(f) ≤ R0(f) ≤ C(f) . In this paper we study a new complexity measure that we call expectational certificate complexity EC(f), which is also a quadratically tight bound on R0(f): EC(f) ≤ R0(f) = O(EC(f) ). We pro...

The space SBV of special BV functions whose gradient measure has no Cantor part was singled out by De Giorgi and Ambrosio [4] as the natural setting to study variational problems where both volume and surface densities have to be taken into account. In fact, for a SBV function f the derivative Df is the sum of a measure Daf absolutely continuous with respect to the Lebesgue measure and a singul...

Recently it has been established that for compact sets F lying on a circle S, the harmonic measure in the complement of F with respect to any point a ∈ S \ F has convex density on any arc of F . In this note we give an alternative proof of this fact which is based on random walks, and which also yields an analogue in higher dimensions: for compact sets F lying on a sphere S in Rn, the harmonic ...

A recent interesting paper by Melton et al. [ 11 discussed finding measures which preserve intuitive orderings on software documents. Informally, if 5 is such an ordering, then they argue that a measure M is a real-valued function defined on documents such that M(F) < M(F')whenever F< F. However, in measurement theory, this is only a necessary condition for a measure M. The representation condi...