Generalized Riemann Integration and an Intrinsic Topology

1980 ◽  
Vol 32 (2) ◽  
pp. 395-413 ◽  
Author(s):  
Ralph Henstock
Keyword(s):  
Compact Space ◽  
Integration Theory ◽  
Locally Compact ◽  
Original Theory ◽  
Locally Compact Space ◽  
Type Integral ◽  
Topological Measure ◽  
Variation Of A Function ◽  
Interesting Theorem ◽  
Riemann Integration

In generalized Riemann integration theory it is becoming increasingly clear that a particular collection of sets has some properties of a topology; it is a useful topology when general requirements hold, and the present paper examines the background. Thomson [23, 24] altered my original theory of the variation and Riemann-type integration that has Lebesgue properties, defining the variation of a function of interval-point pairs over the whole of a space T by using partial divisions of T instead of divisions covering T entirely, and also defining a Lebesgue-type integral. His reason might have been that a decomposable division space seems impossible in a general compact or locally compact space. McGill mentioned this to me, and in [15] connected Thomson's setting with topological measure and Topsøfe [25], giving an interesting theorem on the variation of the limit of a monotone increasing generalized sequence of open sets.

On Unsymmetric Dirichlet Forms

1973 ◽  
Vol 25 (2) ◽  
pp. 252-260 ◽  
Author(s):  
Joanne Elliott
Keyword(s):  
Compact Space ◽  
Measure Space ◽  
Closed Subspace ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Locally Compact ◽  
Locally Compact Space ◽  
Finite Measure Space

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.

Actions That Fiber and Vector Semigroups

1972 ◽  
Vol 24 (1) ◽  
pp. 29-37 ◽  
Author(s):  
T. H. McH. Hanson
Keyword(s):  
Compact Space ◽  
Lie Group ◽  
Fiber Bundle ◽  
Closed Subset ◽  
Locally Compact ◽  
Locally Compact Space

From [2], we can derive a criterion for determining when an action of a Lie group on a locally compact space leads to a fiber bundle. Here, we present an equivalent criterion which can be stated purely in the language of actions of groups on spaces. This is Theorem I. Using this result, we are able to give a version of a result of Home [1] for dimensions greater than one. This is done in Theorem IV and Corollary IVA. In Theorem II, we show that if a vector semigroup acts on a space X, then whenever the map t ↦ tx is 1 — 1 from onto x, it is in fact a homeomorphism. Also, is a closed subset of X. This is also a version of a result in [1].

ON ALGEBRAIC EQUATION WITH COEFFICIENTS FROM THE $ \beta $-UNIFORM ALGEBRA $ C_{\beta} (\Omega) $

2018 ◽  
Vol 52 (3 (247)) ◽  
pp. 161-165
Author(s):  
A.H. Kamalyan ◽  
M.I. Karakhanyan
Keyword(s):  
Algebraic Equation ◽  
Compact Space ◽  
Uniform Algebra ◽  
Locally Compact ◽  
Locally Compact Space

In this work the question of algebraic closeness of $ \beta $-uniform algebra $ A (\Omega) $ defined on locally compact space $ \Omega $ is investigated.

scholarly journals On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)

2015 ◽  
Vol 16 (2) ◽  
pp. 183 ◽  
Author(s):  
O. A. S. Karamzadeh ◽  
M. Namdari ◽  
S. Soltanpour
Keyword(s):  
Prime Ideal ◽  
Compact Space ◽  
Maximal Ideal ◽  
Regular Ring ◽  
Connected Subset ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Space

<p><br />Let $C_c(X)=\{f\in C(X) : |f(X)|\leq \aleph_0\}$, $C^F(X)=\{f\in C(X): |f(X)|&lt;\infty\}$, and $L_c(X)=\{f\in C(X) : \overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq\aleph_0$, and $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). It is shown that if $X$ is a locally compact space, then $L_c(X)=C(X)$ if and only if $X$ is locally scattered.<br />We observe that $L_c(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$.<br /> Spaces $X$ such that $L_c(X)$ is regular (von Neumann) are characterized. Similarly to $C(X)$ and $C_c(X)$, it is shown that $L_c(X)$ is a regular ring if and only if it is $\aleph_0$-selfinjective.<br />We also determine spaces $X$ such that ${\rm Soc}{\big(}L_c(X){\big)}=C_F(X)$ (resp., ${\rm Soc}{\big(}L_c(X){\big)}={\rm Soc}{\big(}C_c(X){\big)}$). It is proved that if $C_F(X)$ is a maximal ideal in $L_c(X)$, then $C_c(X)=C^F(X)=L_c(X)\cong \prod\limits_{i=1}^n R_i$, where $R_i=\mathbb R$ for each $i$, and $X$ has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces $X$ for which $C_F(X)$ is a prime ideal in $L_c(X)$<br />are characterized and consequently for these spaces, we infer that $L_c(X)$ can not be isomorphic to any $C(Y)$. <br /><br /></p>

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