  We study weak convergence of empirical processes of dependent data
$(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially
suitable for data arising from dynamical systems and Markov chains, where the
central limit theorem for partial sums of observables is commonly derived via
the spectral gap technique. We are specifically interested in situations where
the index class ${\mathcal{F}}$ is different from the class of functions $f$
for which we have good properties of the observables $(f(X_i))_{i\geq0}$. We
introduce a new bracketing number to measure the size of the index class
${\mathcal{F}}$ which fits this setting. Our results apply to the empirical
process of data $(X_i)_{i\geq0}$ satisfying a multiple mixing condition. This
includes dynamical systems and Markov chains, if the Perron-Frobenius operator
or the Markov operator has a spectral gap, but also extends beyond this class,
for example, to ergodic torus automorphisms.
