The problem of minimax robust source coding under a fidelity criterion for sources whose statistics belong to uncertainty classes determined by 2-alternating Choquet capacities is examined. We consider (i) single-letter difference distortion criteria for discrete memoryless sources whose probability distributions belong to capacity classes and (ii) the mean-square error distortion criterion for stationary Gaussian sources whose spectral measures belong to capacity classes. Both block source codes and trellis source codes are considered. It is shown that there exists an assemble of block source codes and an ensemble of trellis codes such that for all rates larger than a critical rate and all sources In the class the average distortion converges to any prescribed fidelity level exponentially with Increasing block length or constraint length, respectively. Besides the rate distortion function, the distortion exponent of the class is also evaluated.