The definition of a Gaussian process given by Rasmussen and Williams (RW) in $\textit{Gaussian Processes for Machine Learning}$ is
Definition: $\textit{A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution}$
Comparing this to other sources for example wikipedia "The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.".
I wonder, does "collection" in the definition given by RW refer to an infinite collection since "[...] is the joint distribution of all those (infinitely many) random variables [...] or is a Gaussian process defined for finite collection of random variables?
If Gaussian processes are defined for finite collection of random variable then how can one think of it as "distribution over functions" which is a very common interpretation.