In an effort to develop the foundations for a non-stochastic theory of information, the notion of $delta$-mutual information between uncertain variables is introduced as a generalization of Nairs non-stochastic information functional. Several properties of this new quantity are illustrated, and used to prove a channel coding theorem in a non-stochastic setting. Namely, it is shown that the largest $delta$-mutual information between received and transmitted codewords over $epsilon$-noise channels equals the $(epsilon, delta)$-capacity. This notion of capacity generalizes the Kolmogorov $epsilon$-capacity to packing sets of overlap at most $delta$, and is a variation of a previous definition proposed by one of the authors. Results are then extended to more general noise models, and to non-stochastic, memoryless, stationary channels. Finally, sufficient conditions are established for the factorization of the $delta$-mutual information and to obtain a single letter capacity expression. Compared to previous non-stochastic approaches, the presented theory admits the possibility of decoding errors as in Shannons probabilistic setting, while retaining a worst-case, non-stochastic character.
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