all $f\in {\mathcal{S}}({\mathbb{R}}^d)$. However, it is not obvious that there exists a measure on ${\mathcal{S}}'({\mathbb{R}}^d)$ satisfying these conditions. Since we will rigorously construct the FGF in Section \[Rd\] in the same manner, we will review a construction of white noise following [@simon1979functional].

We say that a complex-valued function $\Phi$ on ${\mathcal{S}}({\mathbb{R}}^d)$ is the characteristic function of a probability measure $\nu$ on ${\mathcal{S}}'({\mathbb{R}}^d)$ if $$\label{eq:char}
    \Phi(\phi) = \int_{{\mathcal{S}}'({\mathbb{R}}^d)}e^{i(x,\phi)}\,d\nu(x),\quad \text{for all }\phi\in
    {\mathcal{S}}({\mathbb{R}}^d).$$

\[minlos\] A complex-valued function $\Phi$ on ${\mathcal{S}}({\mathbb{R}}^d)$ is the characteristic function of a probability measure $\nu$ on ${\mathcal{S}}'({\mathbb{R}}^d)$ if and only if $\Phi(0)=1$, $\Phi$ is continuous, and $\Phi$ is positive definite, that is, $$\sum_{j,k=1}^nz_j\overline{z_k}\Phi(\phi_j-\phi_k)\geq 0,$$ for all $\phi_1,\dots,\phi_n\in {\mathcal{S}}({\mathbb{R}}^d)$, and $z_1,\dots,z_n\in
    \mathbb{C}$. Furthermore, $\Phi$ determines $\nu$ uniquely.

We briefly sketch the proof given in [@simon1979functional Theorem 2.3] for the case $d=1$; the case $d>1$ may be proved similarly. We introduce coordinates to the space ${\mathcal{S}}({\mathbb{R}})$ by writing each function $\phi \in {\mathcal{S}}({\mathbb{R}})$ as $\phi =
    \sum_{n=1}^\infty (\phi,\phi_n)_{L^2({\mathbb{R}})} \phi_n$, where $\{\phi_n\}_{n=0}^\infty$ is the Hermite basis of $L^2({\mathbb{R}})$ defined by $$\phi_n(x) = \frac{(-1)^n e^{\frac{x^2}{2}} \frac{d^n}{d
   x^n}[e^{-x^2}]}{ \pi^{1/4} \sqrt{2^n n!}}.$$ Identifying $\phi \in {\mathcal{S}}({\mathbb{R}})$ with $\{(\phi,\phi_n)_{L^2({\mathbb{R}})}\}_{n=0}^\infty$ and using the fact that $\phi_n$ is an eigenfunction of $-\frac{d^2}{dx^2} + x^2$, we find that 