## Domain Theory and Integration (1995)

Venue: | Theoretical Computer Science |

Citations: | 57 - 12 self |

### BibTeX

@ARTICLE{Edalat95domaintheory,

author = {Abbas Edalat},

title = {Domain Theory and Integration},

journal = {Theoretical Computer Science},

year = {1995},

volume = {151},

pages = {163--193}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a domain-theoretic framework for measure theory and integration of bounded real-valued functions with respect to bounded Borel measures on compact metric spaces. The set of normalised Borel measures of the metric space can be embedded into the maximal elements of the normalised probabilistic power domain of its upper space. Any bounded Borel measure on the compact metric space can then be obtained as the least upper bound of an !-chain of linear combinations of point valuations (simple valuations) on the upper space, thus providing a constructive setup for these measures. We use this setting to define a new notion of integral of a bounded real-valued function with respect to a bounded Borel measure on a compact metric space. By using an !-chain of simple valuations, whose lub is the given Borel measure, we can then obtain increasingly better approximations to the value of the integral, similar to the way the Riemann integral is obtained in calculus by using step functions. ...

### Citations

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Citation Context ...jes integral, but it unfortunately falls short of the constructive features of the Riemann integral. A new idea in measure theory on second countable locally compact Hausdorff spaces was presented in =-=[9]-=-. It was shown that the set of normalised Borel measures on such a space can be embedded into the maximal elements of the probabilistic power domain of its upper space. The image of the embedding cons... |

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Cartesian Closed Categories of Domains, volume 66 of CWI Tracts. Centrum voor Wiskunde en Informatica
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Citation Context ...blished in [9]. We will also present some of the background results, in particular from [17], that we need here. We will use the standard terminology and notations of domain theory, as for example in =-=[18]. Given a dcpo (D; v) a-=-nd a subset A ` D, we let "A = fd 2 D j 9a 2 A: a v dg and " "A = fd 2 D j 9a 2 A: a �� dg where �� is the way-below relation in D. We denote the lattice of open sets of a topol... |

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Citation Context ...lage theorem to find an IFS with contracting affine transformations, whose attractor approximates the image. The theory has many applications including in statistical physics [14, 6, 10], neural nets =-=[5, 8] and image-=- compression [3, 4]. It was shown in [9, Theorem 6.2], that the unique invariant measure �� of an IFS with probabilities as above is the fixed point of the map T : P 1 UX ! P 1 UX �� 7! T (�... |

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