  It is known that the membership in a given reproducing kernel Hilbert space
(RKHS) of the samples of a Gaussian process $X$ is controlled by a certain
nuclear dominance condition. However, it is less clear how to identify a
"small" set of functions (not necessarily a vector space) that contains the
samples. This article presents a general approach for identifying such sets. We
use scaled RKHSs, which can be viewed as a generalisation of Hilbert scales, to
define the sample support set as the largest set which is contained in every
element of full measure under the law of $X$ in the $\sigma$-algebra induced by
the collection of scaled RKHS. This potentially non-measurable set is then
shown to consist of those functions that can be expanded in terms of an
orthonormal basis of the RKHS of the covariance kernel of $X$ and have their
squared basis coefficients bounded away from zero and infinity, a result
suggested by the Karhunen-Lo\`{e}ve theorem.
