Suppose that Ft, t ≥ 0, is an increasing family of σ-fields of subsets of a set Ω, such that ⋂ε > 0 Ft+ε = Ft, for all t ≥ 0.
Let F be a σ-field on Ω such that Ft ⊂ F for all t ≥ 0.
Let P be a probability measure on (Ω, F). Assume that each Ft contains every subset N of Ω included in some set AN ∈ F with P(AN)=0.
Let, for each t ≥ 0, Xt be a measurable function from (Ω, Ft) into a Polish space S, where the Polish space is equipped with the Borel σ-field B, i.e., the smallest σ-field containing all its open sets.
Assume that, for all ω ∈ Ω, the function t → Xt(ω) is continuous or right-continuous with discontinuities of first kind only.
Let B ∈ B, and define TB := inf{t ≥ 0: Xt ∈ B}. Then TB is a measurable function from Ω into [0, ∞] such that {ω ∈ Ω: TB(ω) ≤ t} ∈ Ft, for all t.
Probabilities and Potential, by Claude Dellacherie and Paul-André Meyer.
Let F be a σ-field on Ω such that Ft ⊂ F for all t ≥ 0.
Let P be a probability measure on (Ω, F). Assume that each Ft contains every subset N of Ω included in some set AN ∈ F with P(AN)=0.
Let, for each t ≥ 0, Xt be a measurable function from (Ω, Ft) into a Polish space S, where the Polish space is equipped with the Borel σ-field B, i.e., the smallest σ-field containing all its open sets.
Assume that, for all ω ∈ Ω, the function t → Xt(ω) is continuous or right-continuous with discontinuities of first kind only.
Let B ∈ B, and define TB := inf{t ≥ 0: Xt ∈ B}. Then TB is a measurable function from Ω into [0, ∞] such that {ω ∈ Ω: TB(ω) ≤ t} ∈ Ft, for all t.
Probabilities and Potential, by Claude Dellacherie and Paul-André Meyer.
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