Stopping time property of hitting times

Suppose that F_t, t ≥ 0, is an increasing family of σ-fields of subsets of a set Ω,  such that ⋂_{ε > 0} F_t+ε = F_t, for all t ≥ 0.

Let F be a σ-field on Ω such that F_t ⊂ F for all t ≥ 0.

Let P be a probability measure on (Ω, F). Assume that each F_t contains every subset N of Ω  included in some set A_N ∈ F with P(A_N)=0.

Let, for each t ≥ 0, X_t be a measurable function from (Ω, F_t)  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 → X_t(ω) is continuous or right-continuous with discontinuities of first kind only.

Let B ∈ B, and define T_B := inf{t ≥ 0: X_t ∈ B}. Then T_B is a measurable function from Ω into [0, ∞] such that {ω ∈ Ω: T_B(ω) ≤ t} ∈ F_t, for all t.

Probabilities and Potential, by Claude Dellacherie and Paul-André Meyer.