منابع مشابه
Infinite product measures
The usual proof that the product of a collection of probability measures exists uses Fubini’s theorem. This is unsatisfying because one ought not need to use Fubini’s theorem to prove things having only to do with σ-algebras and measures. In this note I work through the proof given by Saeki of the existence of the product of a collection of probability measures. We speak only about the Lebesgue...
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Let B be the Borel σ-algebra of R, and let B be the Borel σ-algebra of [−∞,∞] = R ∪ {−∞,∞}: the elements of B are those subsets of R of the form B,B ∪ {−∞}, B ∪ {∞}, B ∪ {−∞,∞}, with B ∈ B. Let (X,A , μ) be a measure space. It is a fact that if fn is a sequence of A → B measurable functions then supn fn and infn fn are A → B measurable, and thus if fn is a sequence of A → B measurable functions...
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In this paper, we study cost functions over a finite collection of random variables. For these types of models, a calculus of differentiation is developed that allows us to obtain a closed-form expression for derivatives where “differentiation” has to be understood in the weak sense. The technique for proving the results is new and establishes an interesting link between functional analysis and...
متن کاملImage Measures of Infinite Product Measures and Generalized Bernoulli Convolutions
We examine measure preserving mappings f acting from a probability space (Ω, F, μ) into a probability space (Ω, F , μ) , where μ = μ(f−1). Conditions on f , under which f preserves the relations ”to be singular” and ”to be absolutely continuous” between measures defined on (Ω, F ) and corresponding image measures, are investigated. We apply the results to investigate the distribution of the ran...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1955
ISSN: 0002-9947
DOI: 10.2307/1992844