Reference request: proof that the first hitting time of a Borel set is a stopping time

Question

Where exactly (book and page number) can I find the proof that the first hitting time of a Borel set a "Stopping time" (continuous time). My notes say it is a deep theorem, particularly hard to prove but didn't give any reference. Moreover my prof mentioned that is is so hard to prove that the famous mathematician Laurent Schwarz suggested that one should use this definition of a stopping time/hitting time

Answer 1

The result is known as a Début theorem. Many proofs in the literature use capacities to prove the measurability of the hitting time $$\tau_B := \inf\{t \geq 0; X_t \in B\}$$ (see e.g. Dellacherie & Meyer), but there is also a proof by Bass which is rather elementary. Note that the result is much easier to prove for some particular cases:

1. $(X_t)_t$ is right-continuous and $B$ is open.
2. $(\mathcal{F}_t)_{t \geq 0}$ is complete filtration, $(X_t)_{t \geq 0}$ is a Markov process with càdlàg sample paths and $B$ is closed.

For a proof in the first setting see e.g. Schilling & Partzsch, a proof for the second one you can find in Itô's book (see also this question).

• R.F. Bass: The measurability of hitting times. Electronic communications in Probability 15 (2010), 99-105.
• C. Dellacherie, P.-A. Meyer. Probabilities and Potential
• K. Itô: Stochastic Processes
• R.L. Schilling & L. Partzsch: Brownian Motion