Measure Theory

Halmos, Paul R.

doi:10.1007/978-1-4684-9440-2

Cited by 2,092 publications

(414 citation statements)

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“…It can also be shown (see, for example, Halmos [21], Chapter 10) that each Borel set B contains a sequence of closed subsets C n such that µ(B) = lim n→∞ (µ(C n )).…”

Section: Regular Ks-colouringsmentioning

confidence: 99%

The Bell–Kochen–Specker theorem

Appleby

2005

Studies in History and Philosophy of Science Part B: Studies in

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Meyer, Kent and Clifton (MKC) claim to have nullified the Bell-KochenSpecker (Bell-KS) theorem. It is true that they invalidate KS's account of the theorem's physical implications. However, they do not invalidate Bell's point, that quantum mechanics is inconsistent with the classical assumption, that a measurement tells us about a property previously possessed by the system. This failure of classical ideas about measurement is, perhaps, the single most important implication of quantum mechanics. In a conventional colouring there are some remaining patches of white. MKC fill in these patches, but only at the price of introducing patches where the colouring becomes "pathologically" discontinuous. The discontinuities mean that the colours in these patches are empirically unknowable. We prove a general theorem which shows that their extent is at least as great as the patches of white in a conventional approach. The theorem applies, not only to the MKC colourings, but also to any other such attempt to circumvent the Bell-KS theorem (Pitowsky's colourings, for example). We go on to discuss the implications. MKC do not nullify the Bell-KS theorem. They do, however, show that we did not, hitherto, properly understand the theorem. For that reason their results (and Pitowsky's earlier results) are of major importance.

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“…It can also be shown (see, for example, Halmos [21], Chapter 10) that each Borel set B contains a sequence of closed subsets C n such that µ(B) = lim n→∞ (µ(C n )).…”

Section: Regular Ks-colouringsmentioning

confidence: 99%

The Bell–Kochen–Specker theorem

Appleby

2005

Studies in History and Philosophy of Science Part B: Studies in

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“…x ∈ W s (x). By Lusin's theorem ( [Hl,§55], for example), choose a set U ⊂ X with µ(U ) > 1/2 such that u| U is uniformly continuous.…”

Section: Property (R) and Partial Hyperbolicitymentioning

confidence: 99%

Livsic theorems for connected Lie groups

Pollicott

Walkden

2001

Trans. Amer. Math. Soc.

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Abstract. Let φ be a hyperbolic diffeomorphism on a basic set Λ and let G be a connected Lie group. Let f : Λ → G be Hölder. Assuming that f satisfies a natural partial hyperbolicity assumption, we show that if u : Λ → G is a measurable solution to f = uφ · u −1 a.e., then u must in fact be Hölder. Under an additional centre bunching condition on f , we show that if f assigns 'weight' equal to the identity to each periodic orbit of φ, then f = uφ · u −1 for some Hölder u. These results extend well-known theorems due to Livšic when G is compact or abelian.

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“…The equality (2) uniquely defines a probability measure on the entire σ −algebra B denoted by the same symbol µ (see [13]). Definition 5: [3] A set A C ⊂ X is an attractor for the IFS (1) with respect to the index set C if for all initial condition…”

Section: Dubbini@maildmunipiitmentioning

confidence: 99%

Left invertibility of discrete systems with finite inputs and quantized output

Dubbini

Piccoli

Bicchi

2008

2008 47th IEEE Conference on Decision and Control

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Abstract-The aim of this paper is to address left invertibility for dynamical systems with inputs and outputs in discrete sets. We study systems that evolve in discrete time within a continuous state-space. Quantized outputs are generated by the system according to a given partition of the state-space, while inputs are arbitrary sequences of symbols in a finite alphabet, which are associated to specific actions on the system. We restrict to the case of contractive dynamics for fixed inputs. The problem of left invertibility, i.e. recovering an unknown input sequence from the knowledge of the corresponding output string, is addressed using the theory of Iterated Function Systems (IFS), a tool developed for the study of fractals. We show how the IFS naturally associated to a system and the geometric properties of its attractor are linked to the left invertibility property of the system. Our main results are a necessary and sufficient condition for a given system to be left invertible with probability one on the space of inputs (i.e. for almost all input sequences), and necessary and sufficient conditions for left invertibility and uniform left invertibility under some weak additional hypotheses. A few examples are presented to illustrate the application of the proposed method.

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Measure Theory

Cited by 2,092 publications

References 0 publications

The Bell–Kochen–Specker theorem

The Bell–Kochen–Specker theorem

Livsic theorems for connected Lie groups

Left invertibility of discrete systems with finite inputs and quantized output

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