How does one make sense of a delta function of a scalar field? Disclaimer: Originally posted on math SE, but thought that it was better in physics SE, so deleted my post on math SE and posted here. 
In the classic review summary of stochastic quantization here, in particular equation 3.14, in which the initial condition is given as:
\begin{equation}
P(\phi,0)= \prod\limits_{x} \delta(\phi(x)). \tag{3.14}
\end{equation}
I don't really understand the product because it seems that $\phi(x)$ is a scalar field defined over all of $\mathbb{R}^n$, so I'm not sure how to interpret the product over all $x$. Additionally, what does the delta function mean in this context (that the inputted $\phi = 0$)?
I spent a long time pouring over related literature and couldn't find an explanation of this, so any insight into this would be greatly appreciated.
 A: Ref.1 is already in eq. (3.1) considering a functional integral over the scalar field $\phi:M\to \mathbb{R}$. Here $M$ is spacetime. For a rigorous treatment of functional integrals, Ref. 1 points in the beginning of Section 3 to its Ref. [3.2]. In this answer we will just take an intuitive heuristic approach, and try to construct the functional integral as an appropriate continuum limit of a discretized space-time.
In more detail, if $x_i\in M$ denotes a discrete spacetime point labelled by an index $i\in I$ in a finite index set $I$, then the probability distribution $P$ becomes a (possibly generalized) function 
$$P: ~\mathbb{R}^{|I|}\times \mathbb{R}~\longrightarrow~\mathbb{R} ,$$ 
and eq. (3.14) reduces to a finite product of Dirac distributions
$$P(\phi,t\!=\!0)~=~ \prod\limits_{i\in I} \delta(\phi_i), \qquad \phi_i~:= ~\phi(x_i),\tag {3.14'}$$
which is now well-defined. Here the parameter $t$ is a 'fiducial' time that should not be confused with time in the spacetime $M$. 
Eq. (3.14') becomes the statement that the probability distribution $P$ has 'initial' support at $\phi=0$.
To get finite probabilities one should integrate the probability density $P$ over the variables $\phi_i$, $i\in I$. In other words, ultimately, we want to insert $P$ as an integrand into the functional integral over $\phi$, cf. e.g. eq. (3.20).
References:


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*P.H. Damgaard & H. Huffel, Stochastic Quantization, Phys. Rep. 152 (1987) 227, (pdf).

