Quasi-linear functionals on locally compact spaces

AbstractTopological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an integration over sets yields a new deficient topological measure if we integrate a nonnegative continuous vanishing at infinity function; and it produces a signed deficient topological measure if we integrate a continuous function on a compact space. We present many properties of these resulting deficient topological measures and of signed deficient topological measures. In particular, they are absolutely continuous with respect to the original deficient topological measure, and their corresponding non-linear functionals are Lipschitz continuous. Deficient topological measures obtained by integration over sets can also be obtained from non-linear functionals. We show that for a deficient topological measure μ that assumes finitely many values, there is a function f such that $\begin{array}{} \int\limits_X \end{array}$f dμ = 0, but $\begin{array}{} \int\limits_X \end{array}$ (–f) dμ ≠ 0. We present different criteria for $\begin{array}{} \int\limits_X \end{array}$f dμ = 0. We also prove some convergence results, including a Monotone convergence theorem.

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Integration on locally compact spaces generated by positive linear functionals defined on the spaceof continuous functions with compact supportand the Riesz representation theorem, I

Proceedings of the Japan Academy ◽

10.3792/pja/1195521800 ◽

1966 ◽

Vol 42 (9) ◽

pp. 1033-1037 ◽

Cited By ~ 1

Author(s):

Witold M. Bogdanowicz

Keyword(s):

Representation Theorem ◽

Continuous Functions ◽

Riesz Representation Theorem ◽

Linear Functionals ◽

Locally Compact ◽

Compact Spaces ◽

Positive Linear Functionals ◽

Riesz Representation ◽

Locally Compact Spaces

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On 𝒞ℛ-Epic Embeddings and Absolute 𝒞ℛ-Epic Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2005-043-2 ◽

2005 ◽

Vol 57 (6) ◽

pp. 1121-1138 ◽

Cited By ~ 1

Author(s):

Michael Barr ◽

R. Raphael ◽

R. G. Woods

Keyword(s):

Locally Compact ◽

Compact Spaces ◽

Open Questions ◽

Tychonoff Spaces ◽

Locally Compact Spaces

AbstractWe study Tychonoff spaces X with the property that, for all topological embeddings X → Y, the induced map C(Y ) → C(X) is an epimorphism of rings. Such spaces are called absolute 𝒞ℛ-epic. The simplest examples of absolute 𝒞ℛ-epic spaces are σ-compact locally compact spaces and Lindelöf P-spaces. We show that absolute CR-epic first countable spaces must be locally compact.However, a “bad” class of absolute CR-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute CR-epic abound, and some are presented.

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