This paper shows that the epsilon entropy of any mean-continuous Gaussian process on $L_2\lbrack 0, 1 \rbrack$ is finite for all positive $\epsilon$. The epsilon entropy of such a process is defined as the infimum of the entropies of all partitions of $L_2\lbrack 0, 1 \rbrack$ by measurable sets of diameter at most $\epsilon$, where the probability measure on $L_2$ is the one induced by the process. Fairly tight upper and lower bounds are found as $\epsilon \rightarrow 0$ for the epsilon entropy in terms of the eigenvalues of the process.
Edward C. Posner. Eugene R. Rodemich. Howard Rumsey Jr.. "Epsilon Entropy of Gaussian Processes." Ann. Math. Statist. 40 (4) 1272 - 1296, August, 1969. https://doi.org/10.1214/aoms/1177697502