Lemma 6

Let $\mu \in \mathscr{M}^1$. There exists a sequence of measures $\mu_n \in \mathscr{D}$ such that $\mu(f) = \lim_{n \to \infty} \mu_n(f)$ for all $f \in \mathscr{C}$ and $W(\mu) = \lim_{n \to \infty} W(\mu_n)$.

Let $I = [0, 1]$. The space $\mathscr{M}^1$ of bounded measures on $T = ]0, 1[$ will be identified with the subspace of $\mathscr{M}(I)$ formed by the measures that place no weight at $0$. (3) We equip $\mathscr{M}(I)$ with the vague topology. The mapping

(3) That is, the measures on $I$ that are concentrated on $T = I - \{0\}$.

$t \mapsto \varepsilon_t$ of $I$ into $\mathscr{M}(I)$ is continuous (Ch. III, §1, No. 9, Prop. 13); since D is dense in I, the closure $\overline{\mathscr{D}}$ of $\mathscr{D}$ contains all of the point measures. Let A be the set of measures $\nu \in \mathscr{D}$ such that $\| \nu \| \leq \| \mu \|$; the measure $\mu$ is in the closure of A (Ch. III, §2, No. 4, Cor. 1 of Th. 1). The set A is relatively compact in $\mathscr{M}(I)$ (Ch. III, §1, No. 9, Prop. 15) and the compact subsets of $\mathscr{M}(I)$ are metrizable (TVS, III, §3, No. 4, Cor. 2 of Prop. 6,(4) and GT, X, §3, No. 3, Th. 1). Therefore there exists a sequence of measures $\mu_n \in A$ converging to $\mu$ in $\mathscr{M}(I)$. Since $\mathscr{C}$ is identified with the subspace of continuous functions on I zero at the origin, we have $\mu(f) = \lim_{n \to \infty} \mu_n(f)$ for all $f \in \mathscr{C}$. Moreover, since $\mathscr{C}(I) \otimes \mathscr{C}(I)$ is dense in the normed space $\mathscr{C}(I \times I)$ (Ch. III, §4, No. 1, Lemma 1), the relations $\lim_{n \to \infty} \mu_n = \mu$ and $\| \mu_n \| \leq \| \mu \|$ imply that $\lim_{n \to \infty} (\mu_n \otimes \mu_n) = \mu \otimes \mu$ (Ch. III, §1, No. 10, Prop. 17); since the measures $\mu_n$ and $\mu$ place no weight at 0, we have

whence $\lim_{n \to \infty} W(\mu_n) = W(\mu)$.

It remains to prove that the Fourier transform of $w'$ is equal to $e^{-W/2}$. Let $\mu \in \mathscr{M}^1$; choose measures $\mu_n \in \mathscr{D}$ as in Lemma 6. The measure $w'$ is bounded, and $|e^{i \langle f, \mu_n \rangle}| = 1$ for all $n$; Lemma 5 and Lebesgue's convergence theorem (Ch. IV, §4, No. 3, Th. 2) then imply

Q.E.D.

The measure $w$ on $\mathscr{C}$ whose Fourier transform is equal to $e^{-W/2}$ is called the Wiener measure on $\mathscr{C}$.