$\newcommand{\real}{\mathrm{real}}$I am having trouble with understanding the axiom (OS3) in this book by Glimm and Jaffe. It defines \begin{equation} \mathcal{A} = \left \{ A(\phi) = \sum_{j = 1}^N c_j \exp \left( \phi(f_j) \right) \; \Big \vert \; c_j \in \mathbb{C}, f_j \in \mathcal{D} \right \} \end{equation} where $\mathcal{D}$ is supposed to be the space of compactly supported, smooth functions on $\mathbb{R}^d$ and $\phi \in \mathcal{D}'(\mathbb{R}^d)$. Now, this is notationally somewhat messed up (imho) but it is also stated earlier in the text, that $\mathcal{D}'(\mathbb{R}^d)$ should refer to real distributions hence, I assume that $\mathcal{D}$ should also refer to real functions - but I have not found this statement anywhere.

They then go on by defining the time-reflection operator $\theta : \mathcal{D} \to \mathcal{D}$ and consider a special supspace $\mathcal{A}_+ \subset \mathcal{A}$ and give the axiom: \begin{equation} 0 \le \int_{\mathcal{D}'(\mathbb{R}^d)} \left( \theta A \right)^- A \mathrm{d} \mu \end{equation} where the superscript $-$ is supposed to mean complex conjugation. This might sound okay but then they state that this is equivalent to the positive definiteness of matrices of the form \begin{equation} M_{i j} = \int_{\mathcal{D}'(\mathbb{R}^d)} \exp \left[ i \phi \left( f_i - \theta f_j \right) \right] \mathrm{d} \mu \left( \phi \right) \end{equation} with $f_j$ in the new space $\mathcal{D}_{\real}(\mathbb{R}^d)$.

I feel that it is completely unclear whether they are talking about $\mathcal{D}$ as a space of real or complex functions. Furthermore, I cannot see the equivalence of the two statements unless $\mathcal{A}$ was defined with an additional $i$ in the exponent and with $\mathcal{D} \supset \mathcal{D}_{\real}(\mathbb{R}^d)$ as a space of real functions.

nlab also gives the almost identical definition as the book but without stating anything about real or complex spaces.