6.2 The càdlàg modulus \(w'\)
Billingsley’s modulus \(w'_f(\delta )\) is formalized as the infimum of all \(\varepsilon \ge 0\) admitting a finite partition \(0=t_0{\lt}t_1{\lt}\dots {\lt}t_n=1\) whose cells all have length \({\gt}\delta \) and on each of whose half-open cells \([t_i,t_{i+1})\) the oscillation from the left endpoint value satisfies \(|f(x)-f(t_i)|\le \varepsilon \). Note the nonstandard every-cell convention: every cell (not all but the last) must be longer than \(\delta \); this is harmless for the \(\delta \to 0\) limit, since any finite partition is \(\delta \)-sparse for \(\delta \) below its minimal gap. The modulus is nonnegative and monotone in \(\delta \).
For every càdlàg \(f\) and \(\varepsilon {\gt}0\) there is a finite partition \(0=t_0{\lt}\dots {\lt}t_n=1\) with left-endpoint oscillation \({\lt}\varepsilon \) on every cell (a greedy left-to-right construction; termination uses the existence of left limits). As a consequence, for càdlàg \(f\) the modulus satisfies \(w'_f(\delta )\to 0\) as \(\delta \to 0^+\). This is the technical primitive behind separability, compactness, and the measurability of paths.