Vector-Valued Stochastic Processes. II. A Radon-Nikodym Theorem for Vector-Valued Processes with Finite Variation

1988 ◽  
Vol 102 (2) ◽  
pp. 393
Author(s):  
Nicolae Dinculeanu
Keyword(s):  
Stochastic Processes ◽  
Finite Variation ◽  
Vector Valued ◽  
Nikodym Theorem

scholarly journals Toward a fundamental theorem of quantal measure theory

2012 ◽  
Vol 22 (5) ◽  
pp. 816-852 ◽  
Author(s):  
RAFAEL D. SORKIN
Keyword(s):  
Stochastic Processes ◽  
Path Integral ◽  
Fundamental Theorem ◽  
Measure Theory ◽  
Finite Lattice ◽  
Extension Problem ◽  
Integral Type ◽  
Causal Sets ◽  
Simple Event ◽  
Vector Valued

In this paper we address the extension problem for quantal measures of path-integral type, concentrating on two cases: sequential growth of causal sets and a particle moving on the finite lattice ℤn. In both cases, the dynamics can be coded into a vector-valued measure μ on Ω, the space of all histories. Initially, μ is just defined on special subsets of Ω called cylinder events, and we would like to extend it to a larger family of subsets (events) in analogy to the way this is done in the classical theory of stochastic processes. Since quantally μ is generally not of bounded variation, a new method is required. We propose a method that defines the measure of an event by means of a sequence of simpler events that in a suitable sense converges to the event whose measure we are seeking to define. To this end, we introduce canonical sequences approximating certain events, and we propose a measure-based criterion for the convergence of such sequences. Applying the method, we encounter a simple event whose measure is zero classically but non-zero quantally.

scholarly journals Vector valued mean-periodic functions on groups

2002 ◽  
Vol 72 (3) ◽  
pp. 363-388 ◽  
Author(s):  
P. Devaraj ◽  
Inder K. Rana
Keyword(s):  
Banach Space ◽  
Spectral Analysis ◽  
Spectral Synthesis ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Compact Sets ◽  
Periodic Functions ◽  
Finite Variation ◽  
Locally Compact ◽  
Vector Valued

AbstractLet G be a locally compact Hausdorif abelian group and X be a complex Banach space. Let C(G, X) denote the space of all continuous functions f: G → X, with the topology of uniform convergence on compact sets. Let X′ denote the dual of X with the weak* topology. Let Mc(G, X′) denote the space of all X′-valued compactly supported regular measures of finite variation on G. For a function f ∈ C(G, X) and μ ∈ Mc(G, X′), we define the notion of convolution f * μ. A function f ∈ C(G, X) is called mean-periodic if there exists a non-trivial measure μ ∈ Mc(G, X′) such that f * μ = 0. For μ ∈ Mc(G, X′), let M P(μ) = {f ∈ C(G, X): f * μ = 0} and let M P(G, X) = ∪μ M P(μ). In this paper we analyse the following questions: Is M P(G, X) ≠ 0? Is M P(G, X) ≠ C(G, X)? Is M P(G, X) dense in C(G, X)? Is M P(μ) generated by ‘exponential monomials’ in it? We answer these questions for the groups G = ℝ, the real line, and G = T, the circle group. Problems of spectral analysis and spectral synthesis for C(ℝ, X) and C(T, X) are also analysed.

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