scholarly journals Condensor Principle and the Unit Contraction

1967 ◽  
Vol 30 ◽  
pp. 9-28 ◽  
Author(s):  
Masayuki Itô
Keyword(s):  
Positive Measure ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Dirichlet Space ◽  
Functional Space ◽  
Dirichlet Spaces ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Regular Functional ◽  
Locally Compact Hausdorff Space

Deny introduced in [4] the notion of functional spaces by generalizing Dirichlet spaces. In this paper, we shall give the following necessary and sufficient conditions for a functional space to be a real Dirichlet space.Let be a regular functional space with respect to a locally compact Hausdorff space X and a positive measure ξ in X. The following four conditions are equivalent.

scholarly journals On Total Masses of Balayaged Measures

1967 ◽  
Vol 30 ◽  
pp. 263-278 ◽  
Author(s):  
Masayuki Itô
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dirichlet Space ◽  
Dirichlet Spaces ◽  
Locally Compact ◽  
Equilibrium Measures ◽  
Locally Compact Hausdorff Space

Beurling and Deny [1], [2] introduced the notion of Dirichlet spaces. They [2] showed the existence of balayaged measures and equilibrium measures in the theory of Dirichlet spaces. In this paper, we shall show that the following equivalence is valid for a Dirichlet space on a locally compact Hausdorff space X.

scholarly journals A Riemann integral in a locally compact Hausdorff space

1986 ◽  
Vol 41 (1) ◽  
pp. 115-137 ◽  
Author(s):  
S. I. Ahmed ◽  
W. F. Pfeffer
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Condition ◽  
Lebesgue Integral ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Riemann Integral ◽  
Variational Integrals ◽  
Necessary And Sufficient ◽  
Locally Compact Hausdorff Space

AbstractWe present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

scholarly journals Maximum Principles in the Potential Theory

1963 ◽  
Vol 23 ◽  
pp. 165-187 ◽  
Author(s):  
Masanori Kishi
Keyword(s):  
Maximum Principle ◽  
Potential Theory ◽  
Positive Measure ◽  
Hausdorff Space ◽  
Finite Energy ◽  
Maximum Principles ◽  
Compact Set ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Locally Compact Hausdorff Space

Ninomiya, in his thesis [13] on the potential theory with respect to a positive symmetric continuous kernel G on a locally compact Hausdorff space Ω, proves that G satisfies the balayage (resp. equilibrium) principle if and only if G satisfies the domination (resp. maximum) principle. He starts from the Gauss-Ninomiya variation and shows that for any given compact set K in Ω and any positive upper semi-continuous function u on K, there exists a positive measure μ on K such that its potential Gμ is ≥ u on the support of μ and Gμ≥u on K almost everywhere with respect to any positive measure with finite energy.

Concerning σ-Connectedness of Baire Spaces

1980 ◽  
Vol 32 (6) ◽  
pp. 1438-1447
Author(s):  
A. García-Máynez
Keyword(s):  
Compact Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Compact Sets ◽  
Connected Space ◽  
Locally Compact ◽  
Whole Space ◽  
Locally Compact Hausdorff Space

A well known theorem of Sierpiński states that every compact connected Hausdorff space is σ-connected. Hence, if X is locally compact and Hausdorff and X is locally connected at x, then x has a σ-connected neighborhood. However, local connectedness at x is not a necessary condition for x to have a σ-connected neighborhood, because the whole space may be σ-connected without being locally connected at x. One of the purposes of the present paper is then to investigate which points of a given locally compact Hausdorff space have σ-connected neighborhoods. We find also sufficient conditions for a connected, hereditarily Baire space to be σ-connected and prove the impossibility of expressing a connected, Čech-complete, rim compact space as a countable infinite union of mutually disjoint compact sets. Finally, we introduce the concept of D-connected space and relate it to σ-connectedness.

scholarly journals The Singular Measure of a Dirichlet Space

1968 ◽  
Vol 32 ◽  
pp. 337-359 ◽  
Author(s):  
Masayuki Itô
Keyword(s):  
Compact Hausdorff Space ◽  
Radon Measure ◽  
Hausdorff Space ◽  
Dirichlet Space ◽  
Singular Measure ◽  
Locally Compact ◽  
Positive Radon Measure ◽  
Locally Compact Hausdorff Space

We [4], [5] examined some properties of balayaged measures in the theory of a Dirichlet space. In those papers, we showed that the singular measure of a Dirichlet space plays some important roles. In this paper, we shall precisely examine some properties of the singular measure of a Dirichlet space. Let X be a locally compact Hausdorff space in which there exists a positive Radon measure ξ which is everywhere dense in X.

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

Measures Defined by Gages

1992 ◽  
Vol 44 (6) ◽  
pp. 1303-1316 ◽  
Author(s):  
Washek F. Pfeffer ◽  
Brian S. Thomson
Keyword(s):  
Set Theory ◽  
Measure Space ◽  
Hausdorff Space ◽  
Outer Measure ◽  
Riesz Representation Theorem ◽  
Locally Compact ◽  
Riemann Integral ◽  
Riesz Representation ◽  
Intuitive Concept ◽  
Locally Compact Hausdorff Space

AbstractUsing ideas of McShane ([4, Example 3]), a detailed development of the Riemann integral in a locally compact Hausdorff space X was presented in [1]. There the Riemann integral is derived from a finitely additive volume v defined on a suitable semiring of subsets of X. Vis-à-vis the Riesz representation theorem ([8, Theorem 2.141), the integral generates a Riesz measure v in X, whose relationship to the volume v was carefully investigated in [1, Section 7].In the present paper, we use the same setting as in [1] but produce the measure directly without introducing the Riemann integral. Specifically, we define an outer measure by means of gages and introduce a very intuitive concept of gage measurability that is different from the usual Carathéodory définition. We prove that if the outer measure is σ-finite, the resulting measure space is identical to that defined by means of the Carathéodory technique, and consequently to that of [1, Section 7]. If the outer measure is not σ-finite, we investigate the gage measurability of Carathéodory measurable sets that are σ-finite. Somewhat surprisingly, it turns out that this depends on the axioms of set theory.

scholarly journals On the theorem of Kishi for a continuous function-kernel

1974 ◽  
Vol 53 ◽  
pp. 127-135 ◽  
Author(s):  
Isao Higuchi ◽  
Masayuki Itô
Keyword(s):  
Continuous Function ◽  
Potential Theory ◽  
Compact Hausdorff Space ◽  
Symmetric Function ◽  
Hausdorff Space ◽  
Locally Compact ◽  
Continuity Principle ◽  
Lower Semi ◽  
Locally Compact Hausdorff Space

In the potential theory with respect to a non-symmetric function-kernel, the following theorem is obtained by M. Kishi ([3]).Let X be a locally compact Hausdorff space and G be a lower semi-continuous function-kernel on X. Assume that G(x, x)>0 for any x in X and that G and the adjoint kernel Ğ of G satisfy “the continuity principle”.

scholarly journals More On the countability index and the density index of S(X)

1990 ◽  
Vol 33 (1) ◽  
pp. 159-164
Author(s):  
K. D. Magill
Keyword(s):  
Finite Number ◽  
Topological Semigroup ◽  
Hausdorff Space ◽  
Extension Property ◽  
Dimensional Subspace ◽  
Countable Subset ◽  
Compact Open Topology ◽  
Locally Compact ◽  
Density Index ◽  
Locally Compact Hausdorff Space

The countability index, C(S), of a semigroup S is the smallest integer n, if it exists, such that every countable subset of S is contained in a subsemigroup with n generators. If no such integer exists, define C(S) = ∞. The density index, D(S), of a topological semigroup S is the smallest integer n, if it exists, such that S contains a dense subsemigroup with n generators. If no such integer exists, define D(S) = ∞. S(X) is the topological semigroup of all continuous selfmaps of the locally compact Hausdorff space X where S(X) is given the compact-open topology. Various results are obtained about C(S(X)) and D(S(X)). For example, if X consists of a finite number (< 1) of components, each of which is a compact N-dimensional subspace of Euclidean Nspace and has the internal extension property and X is not the two point discrete space. Then C(S(X)) exceeds two but is finite. There are additional results for C(S(X)) and similar results for D(S(X)).

On Countably Paracompact Normal Spaces

1957 ◽  
Vol 9 ◽  
pp. 443-449 ◽  
Author(s):  
M. J. Mansfield
Keyword(s):  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normal Space ◽  
Countably Paracompact ◽  
Necessary And Sufficient

A. H. Stone (9), E. Michael (3, 4), J. L. Kelley and J. S. Griff en (2) have established many necessary and sufficient conditions that a regular Hausdorff space be paracompact. It is the purpose of this note to show that if the word “countable” is inserted in the appropriate places in the above-mentioned conditions they become, in general, necessary and sufficient conditions that a normal space be countably paracompact.

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