Abstract: Gaussian processes are commonly used tools for modeling continuous processes in machine learning and statistics. This is partly due to the fact that one may employ a Gaussian process as an interpolator for a finite set of known points, which can then be used for prediction and straight forward uncertainty quantification at other locations. However, it is not always the case that the available information is in the form of a finite collection of points. For example, boundary value problems contain information on the boundary of a domain, which is an uncountable collection of points that cannot be incorporated into typical Gaussian process techniques. We propose and construct Gaussian processes that unify, via reproducing kernel Hilbert space, the typical finite case with the case of having uncountable information by exploiting the equivalence of conditional expectation and orthogonal projections. We show existence of the proposed Gaussian process and that it is the limit of a conventional Gaussian process conditioned on an increasing but finite number of points. We illustrate the applicability via numerical examples.