= Solution
A real <Gaussian process> $(Z_t)_{t\in T}$ is a family of <random variables> such that every finite vector $(Z_{t_1},\ldots,Z_{t_d})$ has a <multivariate normal distribution>, allowing degenerate distributions. Equivalently every finite real linear combination is a <Gaussian random variable>. Its mean and <covariance kernel> are $m(t)=\mathbb EZ_t$ and $C(s,t)=\mathbb E[(Z_s-m(s))(Z_t-m(t))]$.
For any finite set of indices, these functions give the whole mean vector and covariance matrix. The <characteristic function> of the vector is
$$
\boxed{\mathbb E\exp\!\left(i\sum_{j=1}^d u_jZ_{t_j}\right)
=\exp\!\left(i\sum_j u_jm(t_j)-\frac12\sum_{j,k}u_ju_kC(t_j,t_k)\right).}
$$
The standard uniqueness theorem for <characteristic functions> therefore determines every <finite-dimensional distribution>, including the singular cases. Cylinder events form a generating class for the <product sigma-algebra> on $\mathbb R^T$ and are closed under finite intersections, so agreement on all cylinder events gives agreement of the process laws by the uniqueness theorem for <probability measures>. Thus the mean and covariance determine the law on this coordinate sigma-algebra even when $T$ is uncountable. This statement does not claim uniqueness on an arbitrary finer sigma-algebra of path properties.
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