P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950. xii+304pp.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....transitions from state SystemFailure does not add up to 1. The original speci cation, though, re ects a certain connection and consistency between the given lower and upper bounds of actual transition probabilities. This connection can be formalized. De nition 10. Let (X) be a sigma algebra [17], a non empty set of subsets of a set X that is closed under the formation of complements and countable unions. 1. A 2 probability measure is a subprobability measure, a function : X) 0; 1] such that ( 0 (17) A [ B) A) B) A and B disjoint) 18) X) 1: 19) 2. Given a ....

....y 6=x (fyg) 1 A : 26) But then (25) follows from (X) 1 and (26) 3. If = 3 , then is a probability measure, based on (19) and (25) Conversely, if is a probability measure, then (21) is ensuring 3 (fxg) fxg) for probability measures satisfy (A) 1 (X n A) for all A 2 (X) [17]. 4.2 Modal Markov chains De nition 11 (Modal Markov chains) A modal Markov chain is a tuple, K = h K ; Act; i; 27) where K is a set of states, Act is a set of actions, and : K Act K I is a function such that for all s 2 K , the sets X s def = fs 0 2 K j a 2 Act; s a ....

P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950.

....results of process algebra operators with respect to abstraction. Section 8 briefly covers a probabilistic view, giving rise to loose Markov chains. Finally, Section 9 provides an outlook on future work. We refer to [1] as a general refer5 ence for domain theoretic concepts and results, [12] for basic notions of measure theory, and [28] as a basic reference to category theory. 2 Categories of relations parametric in views Let us recall the category REL of relations. Its objects are all sets X;Y; Z; for sets X and Y , the morphisms 2 REL(X;Y ) are all subsets of X Theta Y ; ....

P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950.

....has to change the description language, the notion of abstraction, and the verification engine completely; for a notable exception see J. Hillston s work in [8] Such changes not only necessitate the knowledge of sophisticated and computationally expensive concepts, such as measure theory [7] and probabilistic bisimulation [13, 1] but also make it hard to embed the qualitative description into such a quantitative view, or to re interpret quantitative results as qualitative judgments. Ideally, one would like to have a uniform family of such triads with view mediating maps across all ....

P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950.

....2 2 [1; 1] 1 Gamma ffl; 1] 0; ffl] full (c) quantitative, fully specified (d) quantitative, loosely specified Fig. 1. Modeling an unreliable medium [19] 2 Categories of relations parametric in views We refer to [1] as a general reference for domain theorectic concepts and results; see e.g. [16] for basic notions of meausure theory, and to [27] as a basic reference to category theory. Let us recall the category REL of relations. Its objects are all sets X;Y; Z; the morphisms 2 REL(X;Y ) are all subsets of X Theta Y ; the identity morphism for X is id X = f(x; x 0 ) 2 X ....

P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950.

.... to [0; 1] One notices that the type O(X) 0; 1] of topological possibility measures looks like the one for continuous valuations [6,7] Recall that a continuous valuation is nothing but a Scott continuous map 2 [O(X) 0; 1] such that satisfies the usual modular law of measure theory [8]: U [ V ) U V ) U) V ) 2) for all U; V 2 O(X) Clearly, such a property cannot be expected from supmaps of that same type. However, topological possibility measures do satisfy a similar modular law if we replace with , the binary supremum in [0; 1] Any topological ....

P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950.

....reached a level of maturity, that allows more challenging data types. One prominent such development has been the extensive work on the semantics of (exact) real number computation [8] and integration [6] The theory of integration, of course, is based upon the notion of measures (see for example [11]) Traditionally, a measure is defined on a oe algebra of sets. The set theoretic operations of such algebras, set complement, for example, cannot be easily reconciled with topological notions. The usual approach is to take the set of opens O(X) of a topological space and generate its oe algebra ....

....a continuous lattice and 2 P L (X) We know that f defined as Phi : X L op is continuous. If S is any subset of X we may define 1 S = x2S f x 2 S = O2F S O where F S is the filter of all opens O containing S. Note that the definition of 2 resembles that of an outer measure [11]. Proposition 12 Let L op be a continuous lattice and 2 P L (X) For the maps 1 and 2 defined above we have: 1) 1 and 2 are equal, 2) let 0 be the map 1 or 2 ; then 0 extends since 0 O = O for all O 2 O(X) 3) 0 ( x) Phi ) x for all x 2 X, 4) and ....

P. R. Halmos, Measure Theory (D. van Norstrand Company, 1950).

....has to change the description language, the notion of abstraction, and the verification engine completely; for a notable exception see J. Hillston s work in [Hil96] Such changes not only necessitate the knowledge of sophisticated and computationally expensive concepts, such as measure theory [Hal50] and probabilistic bisimulation [Bai96] but also make it hard to embed the qualitative description into such a quantitative view, or to interpret quantitative results as qualitative judgments. Ideally, one would like to have a uniform family of such triads with view mediating maps across all ....

P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950.

....On the other hand, in moving from a qualitative to a quantitative view of a system, one ordinarily has to change the description language, the notion of abstraction, and the verification engine completely. This not only necessitates the knowledge of sophisticated concepts, such as measure theory [Hal50], but is also makes it hard to embed the qualitative description into such a quantitative view, or to interpret quantitative results as qualitative judgments. Ideally, one would like to have a uniform family of such triads with mediating maps across all three dimensions (description, abstraction, ....

P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950.

....has to change the description language, the notion of abstraction, and the verification engine completely; for a notable exception see J. Hillston s work in [Hil96] Such changes not only necessitates the knowledge of sophisticated and computationally expensive concepts, such as measure theory [Hal50] and probabilistic bisimulation [Bai96] but is also makes it hard to embed the qualitative description into such a quantitative view, or to interpret quantitative results as qualitative judgments. Ideally, one would like to have a uniform family of such triads with mediating maps across all ....

P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950.

....to a functor. If f : X Y is a continuous function between topological spaces X and Y , P L (f) P L (X) P L (Y ) is defined as P L (f) O = f Gamma1 (O) for all 2 P L (X) and all O 2 O(Y ) This definition corresponds to those used for valuations [7] and in conventional measure theory [3]. Theorem 7. The map P L ( Delta) is well defined and results in a monotonic functor from TOP to SUP, the category of complete sup lattices and sup maps. When restricted to sober spaces, it is even locally continuous, i.e. Scott continuous in its action on morphisms. Proof. Clearly, P L (X) is a ....

P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950.

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P. R. Halmos. Measure Theory. D. van Norstrand Company, 1950. xii+304pp.

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