Your question has already been asked, and beautifully answered, on the Mathematics SE site:

https://math.stackexchange.com/questions/2297424/extending-a-distribution-over-samples-to-a-distribution-over-functions

It sounds like you're not familiar with the concepts of [Gaussian measures on infinite-dimensional spaces](https://en.wikipedia.org/wiki/Gaussian_measure), linear functionals, pushforward measures, etc. thus I'll try to keep it as simple as possible. 

You already know how to define probabilities over real numbers (random variables) and over vectors (again, random variables, even if we usually call them random vectors). Now we want to introduce a probability measure over an infinite-dimensional vector-space: for example, the space $L^2([0,1])$ of square-integrable functions over $I=[0,1]$. Things get complicated now, because when we defined probability on $\mathbb{R}$ or $\mathbb{R}^n$, we were helped by the fact that the Lebesgue measure is defined on both spaces. However, [there exists no Lebesgue measure over $L^2$](https://en.wikipedia.org/wiki/Infinite-dimensional_Lebesgue_measure)(or any infinite-dimensional Banach space, for that matter). There are various solutions to this conundrum, most of which need a good familiarity with Functional Analysis.

However, there's also a simple "trick" based on the [Kolmogorov extension theorem](https://en.wikipedia.org/wiki/Kolmogorov_extension_theorem), which is basically the way stochastic processes are introduced in most of the probability courses which are not heavily measure-theoretic. Now I'm going to be _very_ hand-wavy and non-rigorous, and limit myself to the case of Gaussian processes. If you want a more general definition, you can read the above answer or look up the Wikipedia link. The Kolmogorov extension theorem, applied to your specific use case, states more or less the following: 

 - suppose that, for each finite set of points $S_n=\{ t_1, \dots ,t_n\} \in I$, $\mathbf{x}_n=(x(t_1),\dots,x(t_n))$ has the [multivariate Gaussian distribution](https://en.wikipedia.org/wiki/Multivariate_normal_distribution)
 - suppose now that for all possible $S_n, S_m\vert S_n\subset S_m $, the corresponding probability distribution functions $f_{S_n}(x_1,\dots,x_n)$ and $f_{S_m}(x_1,\dots,x_{n},x_{n+1},\dots,x_m)$ are _consistent_, i.e., if I integrate $f_{S_m}$ with respect to the variables which are in $S_m$ but not in $S_n$, then the resulting pdf is $f_{S_n}$:

$$ \int_{\mathbb{R}^{n-m+1}}f_{S_m}(x_1,\dots,x_m)\text{d}x_{n+1}\dots \text{d}x_m=f_{S_n}(x_1,\dots,x_n) $$

 - then there exist a stochastic process $X$, i.e., a random variable on the space of functions $L^2$, such that, for each finite set $S_n$, the probability distribution of those $n$ points is multivariate Gaussian.

The actual theorem is widely more general, but I guess this is what you were looking for.