What does it mean for a field to be defined by a measure? In Quantum Physics by Glimm and Jaffe they mention on p. 90 that

The Euclidean fields are defined by a probability measure $d\mu(\phi) = d\mu$ on the space of real distributions. Here $d\mu$ plays the same role as does the Feynman-Kac measure in quantum mechanics

and they go on to introduce a generating functional
$$S\{f\} = \int e^{i\phi(f)}d\mu\tag{6.1.4}$$
that is the inverse Fourier transform of a Borel probability measure $d\mu$ on the space of test functions. Here $\phi$ is the field and $f$ is any test function.
I am a little confused on what this all means. I am familiar with measure theory, but not in this context. How does a (probability) measure define a field? What is the significance of this measure?
As a disclaimer, I am unfamiliar with the Feynman-Kac formula.
 A: I believe this is a common issue in bridging the mathematically rigorous analysis of Glimm and Jaffe and the standard Physics discussion in QFT textbooks. But the basic point is this. Imagine you have some field $\phi(x)$ whose dynamics is governed by an action $S[\phi]$. As an operator on some Hilbert space, you may think that $\phi(x)$ is really defined by its insertions into correlation functions $\langle \phi(x)\cdots \rangle$. This is a pretty common idea for example in Conformal Field Theory.
Now, the thing is that these kinds of correlation functions can be obtained from a generating functional $Z[J]$. Indeed, if you know the correlation functions you can construct $Z[J]$ as $$Z[J]=\sum_{n=0}^\infty \dfrac{1}{n!}\int d x_1\cdots dx_n J(x_1)\cdots J(x_n)\langle \phi(x_1)\cdots \phi(x_n)\rangle,$$
and then you may show that $$\langle \phi(x_1)\cdots \phi(x_n)\rangle =\dfrac{\delta^n Z[J]}{\delta J(x_1)\cdots \delta J(x_n)}\bigg|_{J=0}.$$
If you have more fields you then just need to add more sources. Now one may either formally, or rigorously as done in Glimm & Jaffe, consider a functional Fourier transform of $Z[J]$. Such functional Fourier transform defines a measure. Indeed, the claim is that in a suitable space of distributions, there exists a measure $d\mu(\phi)$ such that $$Z[J]=\int d\mu(\phi) \exp\left[i\int dx \phi(x)J(x)\right].$$
This measure $d\mu(\phi)$ is the "path integral measure", which physicists write as "$\mathfrak{D}\phi e^{-S[\phi]}$". Let me stress that the translation-invariant measure $\mathfrak{D}\phi$ that Physicists employ does not really exist as a rigorous construct in measure theory, because there is no infinite-dimensional Lebesgue measure. Nevertheless, the object "$\mathfrak{D}\phi e^{-S[\phi]}$" can be given meaning, and it is this measure $d\mu(\phi)$.
Maybe the non-trivial step then is in realizing that $d\mu(\phi)$ is the Physicist "$\mathfrak{D}\phi e^{-S[\phi]}$". One way of doing this, that I find quite elegant, is by observing that $Z[J]$ has to obey a functional differential equation known as the Dyson-Schwinger equation. If one formally solves this equation by a functional Fourier transform, one finds out $Z[J]$ in the form above, where $d\mu(\phi)$ is really the known "$\mathfrak{D}\phi e^{-S[\phi]}$". One may then realize that the rigorous version of the story must really be what Glimm & Jaffe are writing down in a logically organized manner. One reference for this is the book by Rivers, "Path Integral Methods in QFT", but if I'm not mistaken, Schwartz also discusses the solution to the Dyson-Schwinger functional equation by functional Fourier transform.
Now we can summarize it like this: the measure $d\mu(\phi)$ is really defined by the action, i.e., the dynamics of the theory. It defines the generating functional $Z[J]$ by means of the construction of the functional Fourier transform of a measure, and such generating functional defines, by functional differentiation, insertions of the field $\phi(x)$ into correlation functions. Since a field is defined by its insertions into correlators, ultimately the measure defines the field.
Finally, let me stress that really constructing $d\mu(\phi)$ is easier said than done. It is known how to be done for free fields and for some interacting cases in lower dimensions. The interacting case in four dimensions is really an open problem. Nevertheless, one can resort to perturbation theory. As soon as $d\mu_0(\phi)$ for the free action $S_0[\phi]$ is known, one may perturbatively build $d\mu(\phi)$ for the interacting action $S[\phi]=S_0[\phi]+gV[\phi]$ by really defining the interacting $Z[J]$ through integration against the free measure $d\mu_0(\phi)$. This is the starting point for Feynman diagrams in QFT.
