  Entropy and differential entropy are important quantities in information
theory. A tractable extension to singular random variables-which are neither
discrete nor continuous-has not been available so far. Here, we present such an
extension for the practically relevant class of integer-dimensional singular
random variables. The proposed entropy definition contains the entropy of
discrete random variables and the differential entropy of continuous random
variables as special cases. We show that it transforms in a natural manner
under Lipschitz functions, and that it is invariant under unitary
transformations. We define joint entropy and conditional entropy for
integer-dimensional singular random variables, and we show that the proposed
entropy conveys useful expressions of the mutual information. As first
applications of our entropy definition, we present a result on the minimal
expected codeword length of quantized integer-dimensional singular sources and
a Shannon lower bound for integer-dimensional singular sources.
