Every coordinate evaluation \(f\mapsto f(t)\) is Borel measurable on \(D\), although it is continuous only at paths continuous at \(t\): the proof exhibits \(f(t)\) as the pointwise limit of the integral averages \(\operatorname {intAvg}_n(t,f)=(n{+}1)\int _t^{t+1/(n+1)}f\), which are \(d^\circ \)-continuous by dominated convergence along Skorokhod time changes. Consequently the rational-coordinate map \(\operatorname {evalRat}\colon D\to (\mathbb {Q}\to \mathbb {R})\) is an injective measurable map between standard Borel spaces, hence a measurable embedding: the Borel \(\sigma \)-algebra of \(D\) coincides with the cylinder \(\sigma \)-algebra. The bridge lemma follows: \(X\colon \Omega \to D\) is measurable iff every coordinate \(\omega \mapsto X(\omega )(t)\) is measurable, and laws on \(D\) are determined by their rational finite-dimensional distributions (ext_of_map_evalRat). (The modulus \(f\mapsto w'_f(\delta )\) itself is not proved measurable — the intended statement survives only inside a comment block — and, by design, nothing below needs it.)

For a process \(X\) and \(s,\varepsilon \), \(\operatorname {crossTime}(X,s,\varepsilon )\) is the first time \(t{\gt}s\) with \(|X_t-X_s|{\gt}\varepsilon \), valued in \(\mathbb {R}\cup \{ \top \} \) (\(\top \) if no crossing occurs); for a right-continuous adapted process it is a stopping time of the right-continuous augmentation \(\mathcal{F}^+\) of the filtration. The iterated crossing sequence of a path \(f\in D\) is \(\operatorname {crossSeq}(\varepsilon ,f)_0=0\) and \(\operatorname {crossSeq}(\varepsilon ,f)_{k+1}=\min (\operatorname {crossTime}(\dots )\! \downharpoonright _1,1)\), clamped to \([0,1]\) and set to \(1\) when no further crossing exists. For a measurable \(D\)-valued random element adapted to a right-continuous filtration (\(\mathcal{F}^+=\mathcal{F}\)), every \(\operatorname {crossSeq}\) iterate is an \(\mathcal{F}\)-stopping time; this is the supply of stopping times fed into Aldous’s criterion.