Lemma 3

*For any $t, t'$ in D,

Note that $\frac{t + t'}{2}$ belongs to D. One knows (No. 6, Example 2) that the family $(X(t))_{t \in D}$ is a basis of the topological dual $\Omega'$ of $\Omega$; therefore there exists a symmetric bilinear form $\widehat{M}$ on $\Omega' \times \Omega'$ characterized by $\widehat{M}(X(t), X(t')) = \inf(t, t')$. By construction, the variance of the Gaussian measure m on $\Omega$ is the quadratic form $\xi \mapsto \widehat{M}(\xi, \xi)$ on $\Omega'$. Set, in particular, an easy calculation yields

(2) Espace mesuré: a locally compact space equipped with a measure (Ch. III, 1st edn., §2, No. 2, p. 52).

By Prop. 6 of No. 5 (formula (16)),

the lemma follows immediately from formulas (40) and (41).

B) Construction of a mapping u of $\Omega$ into $\mathscr{C}$: For every integer $n \geqslant 0$, denote by $E_n$ the subspace of $\mathscr{C}$ formed by the functions that are affine on each of the intervals $\left[ \frac{k-1}{2^n}, \frac{k}{2^n} \right]$ for $1 \leqslant k \leqslant 2^n$. An affine function on a compact interval I of $\mathbf{R}$ attains its bounds at the endpoints of I; consequently,

for $f \in E_n$.

For every function $g \in \Omega$ and every integer $n \geqslant 0$, there exists one and only one function $u_n(g)$ that belongs to $E_n$ and coincides with $g$ at every point of $D_n$; we shall write $T_n g = u_{n+1}(g) - u_n(g)$. Since $D_n$ is finite, the mapping $T_n$ of $\Omega$ into $\mathscr{C}$ is continuous, hence $m$-measurable.