Infinite Measure Spaces Research Articles

On the ergodic theory of non-integrable functions and infinite measure spaces

We prove a Chow-Robbins type result for an ergodic, non-negative SSSP, and a similar result for transformations preserving infinite measure, which implies that for these transformations, no “absolute” version of Hopf's theorem can hold.

Continuous ergodic measures on 𝑅^{∞} have disjoint powers

If μ \mu is an ergodic probability measure on an infinite dimensional linear measure space and if there exists an infinite sequence of measurable linear functional on this space such that all nontrivial linear combinations have continuous distribution under μ \mu , then the convolution powers of μ \mu all live on disjoint sets.

Kolmogorov's Extension Theorem for Infinite Measures

We consider the extension problem of a self-consistent family of infinite measures to a completely additive measure. For probability measures, Kolmogorov's extension theorem assures that the extension is uniquely possible. Our results are as follows: (a) For CT-finite measures, we can reduce the problem to the case of probability measures, so that the extension is uniquely possible. As an application, on an infinite dimensional vector space we can construct such a measure that is invariant both under rotations and homotheties with respect to the origin. It is obtained as the limit of ndimensional measure: Also we shall discuss about the Lorentz invariant measure on an infinite dimensional space. (b) If measures are not a-finite, under the additional condition (EG) in §6, the extension is possible but not unique. We shall mention about the largest and the smallest extension. As an application, we can consider the symbolic representation of a flow {7f} defined on an infinite measure space X, namely constructing an appropriate product space VVR and an appropriate measure on IVR, T$ on X is represented by a shift St on jj/R. w(.)_w(.+t)m

Projections on spaces of invariant functions

Let ( X, ∑, μ) be a measure space and S be a semigroup of measure-preserving transformations T: X → X. In case μ( X) < ∞, Aribaud [1] proved the existence of a positive contractive projection P of L 1( μ) such that for every ƒ ϵ L 1(μ), Pf belongs to the closure C 1(ƒ) in L 1(μ) of the convex hull C(ƒ) of the set { ƒ ○ T:T ϵ S}. In this paper we extend this result in three directions: we consider infinite measure spaces, vector-valued functions, and L p spaces with 1 ⩽ p < ∞, and prove that P is in fact the conditional expectation with respect to the σ-algebra Λ of sets of ∑ which are invariant with respect to all T ϵ S.

On decomposition of transformations in infinite measure spaces

On decomposition of transformations in infinite measure spaces

Some representations of the automorphism group of an infinite continuous homogeneous measure algebra

Let (X, r, m) be an infinite continuous homogeneous measure space. Let A be the measure algebra of (X, τ, m) and G be the automorphism group of A. The canonical representation of G on the subspace of all elements of ®nL\X, r, m) of some fixed maximal symmetry type is irreducible. Two such representations are equivalent iff they correspond to the same ne N and to the same partition of n. l Introdution and notation* Let H be a Hubert space. If neN, let (gΓ H denote the tensor product of H with itself n times. Let Sn be the symmetric group on the first n natural numbers. Let Θ be the unique representation of Sn on ®w H such that θ(g)(v1 (x) v2 (x) • ® O = vgω (g) vgW (x) (x) ^(Λ) if geS n and ^.e If for 1 ^ ί ^ w. If is a set of operators on a Hubert space and contains the adjoint of each of its members, then the commutant S' of is a von Neumann algebra; the double commutant S of is the smallest von Neumann algebra which contains [1]. Θ(Sn) is generated by its mutually orthogonal minimal projections [9]. These projections are in a 1-1 correspondence with the maximal symmetry types. The vectors in the range of a minimal projection are said to be of the corresponding maximal symmetry type. Let U{H) be the unitary group on H. Let I be the canonical representation of U(H) on H. If ne N, let <gΓ / be the tensor product of I with itself n times. The restriction of ® % I to the subspace of all vectors of any fixed maximal symmetry type is irreducible. If H is finite dimensional, this result is classical [9]; if H is infinite dimensional, it is due to Segal [8]. The author has obtained similar results [3] for the symmetric groups on an infinite set S, where is an orthonormal basis for H. In all three cases, these representations can be explicitly characterized. Below we prove that the analogous representations of the automorphism group of an infinite homogeneous measure algebra are irreducible. No characterization of these representations is known.

On category of mixing in infinite measure spaces

On category of mixing in infinite measure spaces

Esistenza e condizioni necessarie per controlli ottimi di sistemi definiti da equazioni integrali

In the first two sections of this paper existence theorems are proved for optimal control problems described either by Fredholm systems of integral equations (with control appearing nonlinearly) on a given (possibly infinite) measure space, eventually in presence of eigenvalues, or by Urysohn systems, both with pointwise and integral constraints on controls and states. Controllability is assumed throughout the paper. In the last section necessary conditions are proved for optimal controls of Fredholm systems (a minimum principle in pointwise, or « differential » form), and theorems (that seem to be new also for the corresponding ordinary differential control systems) are deduced about bang-bang and regularization properties of the optimal control (s) (which turns out to be a continuous function except for a finite number of points, if positivity and semicontinuity assumptions hold about the data): in some particular cases the optimal control will be piecewise constant. All these results are independent on the known ones.

Conditional expectations for general measure spaces

The classical conditional expectation with respect to σ-algebras, on probability measure spaces, has been extended for infinite measure spaces. In this paper we consider conditional expectations with respect to δ-rings, on arbitrary measure spaces. An additional condition has to be imposed in order to insure the uniqueness of the conditional expectation. The existence of the conditional expectation is proved for functions in L p with 1 ≤ p < ∞, and, for localizable measures, also in L ∞. The properties of the classical conditional expectation remain true in the general case, sometimes with some modifications.

On mixing in infinite measure spaces

Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote σ-algebra (=tail σ-field). A transformation T is called mixing if (T −n A) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures π1,π2, $$\pi _1 ,\pi _2 ,\pi _1 \circ T^{ - n} - \pi _2 \circ T^{ - n} \to 0$$ in total variation norm.

On the entropy of a product endomorphism in infinite measure spaces

In this note, a relationship is established between the entropy, defined by Krengel for an endomorphism of a σ-finite measure space, and the notion of a spreading partition. This relationship is used to answer in the “quasi-finite” case a question raised by Krengel concerning the entropy of the product endomorphism on the direct product of a finite and σ-finite measure space.

On mixing in infinite measure spaces

Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote σ-algebra (=tail σ-field). A transformation T is called mixing if (T−nA) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures π1,π2, \(\pi _1 ,\pi _2 ,\pi _1 \circ T^{ - n} - \pi _2 \circ T^{ - n} \to 0\) in total variation norm.

On Ergodic Measure-Preserving Transformations Defined on an Infinite Measure Space

On Ergodic Measure-Preserving Transformations Defined on an Infinite Measure Space

On ergodic measure-preserving transformations defined on an infinite measure space

Few, if any, of the properties enjoyed by ergodic measure-preserving transformations defined on a finite measure space generalize in a natural way to those defined on an infinite measure space. Concrete examples of ergodic transformations which preserve a finite measure and ones which preserve an infinite measure exist in the literature, see [1]. It is not difficult to see that ergodic transformations never admit wandering sets of positive measure. In [2 ] it was shown that a basic difference exists between ergodic transformations which preserve a finite measure and those which preserve an infinite measure; namely, an ergodic measure-preserving transformation defined on an infinite measure space always admits weakly wandering sets of positive measure (Theorem 2 of [2]). Unlike wandering sets, it is not true in general that the union of two weakly wandering sets is again a weakly wandering set even if we require that a class of mutually disjoint images of one weakly wandering set does not intersect a class of mutually disjoint images of the other. One may ask then if there are any ergodic measure-preserving transformations defined on an infinite measure space which admit only weakly wandering sets of finite measure. In this paper we show that this is not the case. We prove that there always exist weakly wandering sets of infinite measure for any ergodic measure-preserving transformation defined on an infinite measure space (Theorem 3). In [2] the existence of a weakly wandering set of positive measure for an ergodic measure-preserving transformation defined on an infinite measure space was discovered while studying the necessary and sufficient conditions for the existence of a finite, invariant, and equivalent measure for a given measurable and nonsingular transformation. In this paper we construct the weakly wandering sets in a different way. Using the pointwise ergodic theorem we prove a simple and yet a useful fact about ergodic measure-preserving transformations defined on an infinite measure space (Theorem 2). It states that given two sets A and B both of finite measure, it is possible to find an image of A under some power of the transformation T which has small intersection with the set B. This fact is basic in proving Lemma 1 which shows the existence of a

Notes on infinite product measure spaces, II

Notes on infinite product measure spaces, II

Notes on infinite product measure spaces, I

Notes on infinite product measure spaces, I