A note on higher integrability of projections

CHAPTER III - REGULAR MEASURES

departmental bulletin paper

The novel concept of focality is introduced for Borel probability measures on compact Hausdorff topological spaces. We characterize focal Borel probability measures as those Borel probability measures that are strictly positive on every nonempty open subset. We also prove the existence of focal Borel probability measures on compact metric spaces. Lastly, we prove that the set of focal (regular) Borel probability measures is convex but not extremal in the set of all (regular) Borel probability measures.

1. Introduction. The problems treated in this paper derive from the viewpoint of measure and integration developed in the book of P. R. Halmos [4]. We are concerned, above all, with the formulation of Fubini's theorem in the product of two locally compact spaces, assuming we are given a Borel measure on each of the factor spaces. Our basic tools, treated elegantly in [4], are (1) the theory of the product of two a-finite measure spaces, and (2) the theory of a single Borel measure on a locally compact space. But these tools alone fail to yield a satisfactory Fubini theorem in the context of locally compact spaces. The reason for this failure is that the domain of definition of the product of two Borel measures, as defined in [4], may not be large enough. (Examples are given in ?7 to illustrate insufficient domain. However, a case is given in ?8 in which the domain is sufficient.) To explain this circumstance in greater detail, let us introduce some notations. For the rest of the paper, p and v denote Borel measures on the locally compact spaces X and Y, respectively. (At times we shall assume that ,u or v is regular, or that X = Y.) For the definitions of Borel measure and regular Borel measure, the reader is referred to [4, Chapter X]. Specifically, the Borel sets of X, Y, and X x Y are the a-ring generated by the compact subsets of X, Y, and X x Y, respectively; we denote this class by X(X), X(Y), and X(X x Y), respectively. By the product a-ring of I(X) and X(Y), denoted

In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of type‐2 theory of effectivity. This gives rise to a natural representation of the set of Borel probability measures on X. We compare this representation to a canonically constructed representation which encodes a Borel probability measure as a lower semicontinuous function from the open sets to the unit interval. We show that this canonical representation is admissible with respect to the weak topology on Borel probability measures. Moreover, we prove that for countably‐based topological spaces the representation via probabilistic processes is equivalent to the canonical representation and thus admissible with respect to the weak topology. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

<!-- *** Custom HTML *** --> This chapter deals with the special features of measure theory when the setting is a locally compact Hausdorff space and when the measurable sets are the Borel sets, those generated by the compact sets. Sections 1–2 establish the basic theorem, the Riesz Representation Theorem, which says that any positive linear functional on the space $C_{\mathrm{com}}(X)$ of continuous scalar-valued functions of compact support on the underlying space $X$ is given by integration with respect to a unique Borel measure having a property called regularity. The steps in the construction of the measure run completely parallel to those for Lebesgue measure if one regards the geometric information about lengths of intervals as being encoded in the Riemann integral. The Extension Theorem of Chapter V is the main technical tool. Section 3 studies more closely the nature of regularity of Borel measures. One direct generalization of a Euclidean theorem is that the space of continuous functions of compact support in an open set is dense in every $L^p$ space on that open set for $1\leq p \lt \infty$. A new result is the Helly–Bray Theorem—that any sequence of Borel measures of bounded total measure in a locally compact separable metric space has a weak-star convergent subsequence whose limit is a Borel measure. Section 4 regards $C_{\mathrm{com}}(X)$ as a normed linear space under the supremum norm and identifies the space of continuous linear functionals, with its norm, as a space of signed or complex Borel measures with a regularity property, the norm being the total-variation norm for the signed or complex Borel measure.

Let (X, Bx) be a Blackwell space, where BX is the a-algebra of Borel sets. Then if a is a finite measure defined on a countably generated sub-a-algebra BG Bx, a can be extended to a Borel measure T. Equivalently, if X and Y are Blackwell andf: X-* Y is Borel, and It is a Borel measure carried on f(X)c Y, then there exists a Borel measure T on X with Tf =o, where T'(E)=T(f1 (E)). We characterize {TI Tf = a} if f is semischlicht. Let BX denote the Borel sets of a topological space X. We consider the following measure extension (or equivalently restriction) problem: given a measure (we will always mean finite measure) a defined on a or-algebra Bc Bx, can or be extended to all of Bx, i.e., does there exist a Borel measure isuch that i-(E)=a(E) for all E e B? It is well known (see [1, p. 71], for details) that if B1 and B2 are c-algebras, and B2 is generated by B1 and finitely many additional sets, then any measure on B1 can be extended to B2. The result is not known for countably generated extensions. We show below (Theorem 5) that if X is a Blackwell space and B is a countably generated sub-a-algebra of Bx, then any measure on B extends to Bx. A Blackwell space is a measure space (X, Bx), where X is an analytic subset of a complete separable metric space (c.s.m.). A subset A of a c.s.m. is analytic iff A is the continuous image of a c.s.m. We note that the analytic sets form a proper subset of Ux, the set of absolutely measurable subsets of X, where E e Ux iff E is f-measurable for all finite Borel measures 4u, where '7 denotes the completion of It, i.e., given j, there exist El, E2-e Bx such that El c Ec E2 and M(E2-E1)=O. A function g is said to be absolutely measurable if g-1(V) E Ux for all open V. Details may be found in [3], [4], or [5]. We note that if Xc S, X analytic, S a c.s.m., then Bx={Er)XjEceBS}, so elements of BX are topologically analytic, and not necessarily Borel in S. We begin by considering a special class of sub-cr-algebras of Bx. Let f:X-Y be Borel measurable, and let Bf={f-l(E)JE e Bx}. Given a Borel Presented to the Society, April 20, 1973; received by the editors January 30, 1973 and, in revised form, June 4, 1973. AMS (MOS) subject classifications (1970). Primary 28A05, 28A60, 28-00. 1 Partially supported by NSF GP-38265. ? American Mathematical Society 1974

This chapter presents a study of a finite Borel measure on ℝn and derives the implied properties on an associated function F that generalize &quot;increasing and right continuous&quot; in the 1-dimensional case. It then generalizes this development to general Borel measures, and shows that any function with these properties induces a Borel measure. In this 1-dimensional case, unbounded intervals were manageable in the sense that there were only two types, and such an intervals either had finite measure or not. For Lebesgue and general measures, the process is more subtle. The approach is axiomatic in that one explicitly defines the value of the integral for certain special functions, and then proceeds to prove that this definition extends to a wide class of functions of interest. One also defines the integral to have an important property with respect to this special class of functions, again with the goal to prove that this property extends to the wider class.

K. Fukaya introduced in [11] a topology on a set of metric spaces equipped with Borel measures, called the measured Hausdorff topology and discussed the continuity of the eigenvalues of the Laplace operators of Riemannian manifolds with uniformly bounded curvature. The purpose of the present paper is to study more closely the Laplace operators of Riemannian manifolds which collapse in this topology to a space of lower dimension while keeping their curvature bounded. 0.1. According to [16], a map h:X—>Y of metric spaces is said to be an £-Hausdorff approximation if |dis(#, x')—dis(A(#),h(x')) ^£ for all x,x' ^X and the ^-neighborhood of the image h(X) coincides with Y. A sequence of compact metric spaces {Xf} converges, by definition, to a compact metric space Y in the Hausdorff distance if there are a sequence of positive numbers {£(&')} going to zero as i tends to infinity and £(/)-Hausdorff approximations hi: Xi~+ Y of Xi into Y. Moreovre when each metric space Xs is equipped with a Borel measure μi of unit mass, according to [11], we say that {(Xg, μ, )} converges to Y with a Borel measure μ^ of unit mass in the measured Hausdorff topology, if in addition, these maps kg: Xg-* Y are Borel measurable and the push-forward measure h^μ{ converges to μ^ in the weak* topology. Now we shall consider a sequence of compact Riemannian manifolds {(Mgygg)} of dimension m whose sectional curvature KM. is bounded uniformly in its absolute value by a constant, say 1, and assume that tihs sequence converges to a compact metric space Moo in the Hausdorff distance. When the volume of Mf is bounded uniformly away from zero by a positive constant, Gromov's

Let G be a separable locally compact group (separable in the sense that the topology of G has a countable base). Let S be a standard Borel space on which G acts on the right such that:(1) s · g1g2 = (s · g1) · g2;(2) s · e = s;(3) (s, g) → s · g is a Borel function from S × G to S.If μ is a Borel measure on S, let μg be the Borel measure on S defined by μg(E) = μ(E · g).Let μ be a Borel measure on S which is quasi-invariant under the action of G; i.e., μg and μ are absolutely continuous (g ∈ G). The triple (G, S, μ) is called a dynamical system [11; 8].Consider the following general problem. Let (G, S, μ) be a dynamical system.

This paper provides first tools for generalizing the theory of orthogonal rational functions on the unit circle 𝕋 created by Bultheel, González‐Vera, Hendriksen and Njåstad to the matrix case. A crucial part in this generalization is the definition of the spaces of matrix‐valued rational functions for which an orthogonal basis is to be constructed. An important feature of the matrix case is that these spaces will be considered simultaneously as left and right modules over the algebra ℂq×q. In this modules we will define simultaneously left and right matrix‐valued inner products with the aid of a nonnegative Hermitian‐valued q × q Borel measure on the unit circle. Given a sequence (αj)j∈ℕ of complex numbers located in ℂ\𝕋 (especially in “good position” with respect to the unit circle) we will introduce a concept of rank for nonnegative Hermitian‐valued q × q Borel measures on the unit circle which is based on the Gramian matrix of particular rational matrix‐valued functions with prescribed pole structure. A main result of this paper is that this concept of rank is universal. More precisely, it turns out that the rank of a matrix measure does not depend on the given sequence (αj)j∈ℕ. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

Continuum-wise expansive measures

Measure N-expansive systems

Admissible Representations of Probability Measures