Just repeating my answer of the crosspost: https://math.stackexchange.com/questions/4198282/a-condition-for-the-vacuum-generating-functional

According to the article (first paragraph of Sec. 4) the logical order of things is as follows: you define $C(z)$ (and thus $R(z)$), as well as $E$, then you let *by definition* $\mu(z):=E[e^{iR(z)}]$ (Eq. I). Then you prove that an explicit formula for $\mu(z)$ is $\mu(z)=e^{-\frac{1}{4}|z|^2}$ (Eq. II). The reason $\mu$ satisfies the positivity of Theorem 1 is because of (Eq. I) and not because of (Eq. II). The explanation for positivity via (Eq. I) is the "obvious part" top of page 9 of the article. It is possible to prove positivity from (Eq. II), but that uses the Schur Product Theorem.
This is, for instance, done in the book on quantum physics by Glimm and Jaffe (Theorem 6.2.2 in 1987 Edition). If the positivity condition seems hard for you to fathom, then look up Bochner's Theorem and the notion of functions of positive type, for more background. Finally: not sure this article by Segal is what you want to read.