Background: Topological Data Analysis (TDA) captures multi-scale geometric features of data as persistence diagrams, yet no principled information-theoretic framework quantifies how much information those features carry, how efficiently they compress, or when they are informationally irreducible. Methods: We construct a measure-theoretic probability space over persistence diagram space using a Poisson-process reference measure, and define topological entropy (H-T), topological mutual information (I-T), and a topological rate–distortion function as the core objects of a new theory. Results: Four theorems with full proofs establish finite stability, axiomatic uniqueness, a Topological Data Processing Inequality, and a Rate–Distortion Theorem with explicit Poisson-model closed-form formula. A Renyi generalization of topological entropy is also established. Computational and practical implementation aspects—including finite-sample estimation, multi-parameter extension, and algorithmic realization—are addressed inline throughout the paper. Conclusions: This framework provides a rigorous measure-theoretic information-theoretic foundation for persistent homology, demonstrated on simulated brain connectivity and point cloud data, with applications to threshold selection, genomic classification bounds, and compressed sensing.
Perumalsamy et al. (Mon,) studied this question.