Abstract
(a) Filtrations of a Measurable Space. Let (Ω, ℱ) be a measurable space, and let (I, ≤) be a linearly ordered set. A filtration of (Ω, ℱ) is a map t↦ℱ(t) from I into the class of sub σ algebras of ℱ, increasing in the sense that If (Ω, ℱ, ℱ(•)) is a filtered measurable space and if the index set I is an interval in ℝ ordered by ≤, define ℱ+(t)= ⋂ s>t ℱ(s) for tin I to get a filtration ℱ+(•) with ℝ (t)⊂ℱ+(t) for t in I. The filtration ℱ(•) is said to be right continous if ℱ(•)=ℱ+(•). In particular, ℱ+(•) is necessarily right continous.
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© 1984 Springer-Verlag New York Inc.
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Doob, J.L. (1984). Fundamental Concepts of Probability. In: Classical Potential Theory and Its Probabilistic Counterpart. Grundlehren der mathematischen Wissenschaften, vol 262. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5208-5_20
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DOI: https://doi.org/10.1007/978-1-4612-5208-5_20
Publisher Name: Springer, New York, NY
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