An Approach of Randomness of a Sample Based on Its Weak Ergodic Limit

For a Polish Sample Space with a Borel σ-field with a surjective measurable transformation, we define an equivalence relation on sample points according to their ergodic limiting averages. We show that this equivalence relation partitions the subset of sample points on measurable invariant subsets, where each limiting distribution is the unique ergodic probability measure defined on each set. The results obtained suggest some natural objects for the model of a probabilistic time-invariant phenomenon are uniquely ergodic probability spaces. As a consequence of the results gained in this paper, we propose a notion of randomness that is weaker than recent approaches to Schnorr randomness.

The Investigation of a Novel Large-Strain Sensor Based on a Dual Capacitive Structure

In this paper, a novel large-strain sensor based on a dual planar capacitive structure has been developed. It has the capacity of large-strain measurement up to 200,000 με (0.2 ε). The change in strain causes a measurable transformation in the capacitance of the sensor by relative shift of the overlap area between two capacitive plates, one fixed (i.e. fixed plate) and the other one movable (i.e. movable plate), and is thus converted into a voltage signal by a read-out circuit module. The dual capacitor structure was designed for increasing the initial capacitance and improving the resolution of sensors compared with a single capacitor structure. The experimental results showed that the sensor had a linearity of 2.29% full scale (FS), a hysteresis error of 1.146%FS, repeatability of 0.226%FS and a resolution of 0.5%FS, suggesting excellent performance of the sensor.

Injective composition operators on Lorentz–Bochner spaces

In this paper, we extend the notion of essential range to vector-valued functions and present various equivalent conditions for the injectiveness of the composition operators alongwith a characterisation for measurable transformations inducing composition operators between Lorentz–Bochner spaces. Some aspects of the weighted composition operators on Lorentz–Bochner spaces, induced by a measurable transformation and an operator valued map, are also discussed.

Dynamics via measurability

Generatorsfforσ-algebras can be used to view the dynamics of an invertible measurable transformationTin terms of the range values off∘T. Such generators are the norm rather than the exception. Related measurable and quantitative methods of estimating a function from the behavior of ergodic averages are also discussed.

Weighted inequalities for Cesàro maximal operators in Orlicz spaces

Let 0 < α ≤ 1 and let M+α be the Cesàro maximal operator of order α defined by In this work we characterize the pairs of measurable, positive and locally integrable functions (u, v) for which there exists a constant C > 0 such that the inequality holds for all λ > 0 and every f in the Orlicz space LΦ(v). We also characterize the measurable, positive and locally integrable functions w such that the integral inequality holds for every f ∈ LΦ(w). The discrete versions of this results allow us, by techniques of transference, to prove weighted inequalities for the Cesàro maximal ergodic operator associated with an invertible measurable transformation, T, which preserves the measure.Finally, we give sufficient conditions on w for the convergence of the sequence of Cesàro-α ergodic averages for all functions in the weighted Orlicz space LΦ(w).

On the almost everywhere convergence of the ergodic averages

Let (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

The Pointwise Ergodic Theorem for Transformations whose Orbits contain or are contained in the Orbits of a Measure-Preserving Transformation

1. Introduction. Let be a probability space with standard. Let T be a bimeasurable one-to-one map of Ω onto itself. Let U: Ω → Ω be another measurable transformation whose orbits are contained in the T-orbits; that is,where Z denotes the set of integers. (This is equivalent to saying that there is a measurable mapping L: Ω → Z such that U(ω) = TL(ω)(ω), ω ∈ Ω.) Such pairs (T, U) arise quite naturally in ergodic theory and information theory. (For example, in ergodic theory, one can see such pairs in the study of the full group of a transformation [1]; in information theory, such a pair can be associated with the input and output of a variable-length source encoder [2] [3].) Neveu [4] obtained necessary and sufficient conditions for U to be measure-preserving if T is measure-preserving. However, if U fails to be measure-preserving, U might still possess many of the features of measure-preserving transformations.

Compact composition operators

Let (Хζ,λ) be a σ-finite measure space, and let ϕ be a non-singular measurable transformation from X into itself. Then a composition transformation Cϕ on L2(λ) is defined by Cϕf = f ∘ ϕ. If Cϕ is a bounded operator, then it is called a composition operator. The space L2(λ) is said to admit compact composition operators if there exists a ϕ such that Cϕ is compact. This note is a report on the spaces which admit or which do not admit compact composition operators.