We develop a framework for rate-distortion theory and a theory of
quantization for (sequences of) random variables of general
distribution supported on general sets including manifolds and fractal
sets. Manifold structures are prevalent in data science, e.g., in
compressed sensing, machine learning, image processing, and
handwritten digit recognition. Fractal sets find application in image
compression and in the modeling of Ethernet traffic. Our contribution
in rate-distortion theory is the derivation of a lower bound on the
rate-distortion function that applies to random variables of general
distribution and for continuous random variables reduces to the
classical Shannon lower bound. The only requirement for our lower
bound to apply is that the distribution of the random variable is
absolutely continuous with respect to a sigma-finite measure of finite
generalized entropy satisfying a certain subregularity condition. This
condition is very general and prevents the measure from being highly
concentrated on balls of small radii. Our contribution in quantization
is the derivation of a lower bound on the n-th quantization error for
random variables where the distribution is absolutely continuous with
respect to a finite measure satisfying the above mentioned
subregularity condition. To illustrate the wide applicability of our
results, we evaluate these lower bounds for a random variable
distributed uniformly on a manifold, namely, the unit circle, and a
random variable distributed uniformly on a self-similar set, namely,
the middle third Cantor set.