A Lévy noise on Rd assigns a random real “mass” Π(B) to each Borel subset B of Rd with finite Lebesgue measure. The distribution of Π(B) only depends on the Lebesgue measure of B, and if B1, . . . , Bn is a finite collection of pairwise disjoint sets, then the random variables Π(B1), . . . , Π(Bn) are independent with Π(B1 ∪ · · · ∪Bn) = Π(B1) + · · ·+ Π(Bn) almost surely. In particular, the di...

*Correspondence: [email protected] College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China Abstract The main goal of the paper is to establish the boundedness of the fractional type Marcinkiewicz integralMβ ,ρ ,q on non-homogeneous metric measure space which includes the upper doubling and the geometrically doubling conditions. Under the a...

These are some brief notes on measure theory, concentrating on Lebesgue measure on Rn. Some missing topics I would have liked to have included had time permitted are: the change of variable formula for the Lebesgue integral on Rn; absolutely continuous functions and functions of bounded variation of a single variable and their connection with Lebesgue-Stieltjes measures on R; Radon measures on ...

• Rn has a natural measure space structure; namely, Lebesgue measure m on the Borel σalgebra. The most important property of Lebesgue measure is that it is invariant under translation. This leads to nice interactions between differentiation and integration, such as integration by parts, and it gives nice functional-analytic properties to differentiation operators: for instance, the Laplacian ∆ ...

It is proved that for any d ≥ 3, there exists a norm ‖ · ‖ and two points a, b in Rd such that the boundary of the Leibniz half-space H(a, b) = {x ∈ Rd : ‖x − a‖ ≤ ‖x − b‖} has non-zero Lebesgue measure. When d = 2, it is known that the boundary must have zero Lebesgue measure.

We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schrödinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero Lebesgue measure. No example with those properties was previously known, even in the larger class of ergodic potentials. We also conclude that the generic limit periodic potential has a spectrum of zero Lebe...

For a sequence of subadditive potentials, a method of choosing state points with negative growth rates for an ergodic dynamical system was given in [5]. This paper first generalizes this result to the non-ergodic dynamics, and then proves that under some mild additional hypothesis, one can choose points with negative growth rates from a positive Lebesgue measure set, even if the system does not...

We denote by x a real variable and by n a positive integer variable. The reference measure on the real line R is the Lebesgue measure. In this note we will use only basic properties of the Lebesgue measure and integral on R. It is well known that the fact that a function tends to zero at infinity is a condition neither necessary nor sufficient for this function to be integrable. However, we hav...

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...