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

Approximating class approach for empirical processes of dependent sequences indexed by functions