Evaluation of $\varepsilon$-entropy of random variables for small $\varepsilon$

An expression for the $\varepsilon$-entropy $H_\varepsilon(\xi)$ is derived for an $n$-dimensional random variable $\xi$ whose density distribution satisfies rather general conditions. An asymptotic expression for $H_\varepsilon(\xi)$ as $\varepsilon\to0$ is obtained for the case when the accuracy of transmission is given by a loss function $\rho(x,y)$ satisfying some weak conditions. Specific cases are examined.