  Conceptually, partial information decomposition (PID) is concerned with
separating the information contributions several sources hold about a certain
target by decomposing the corresponding joint mutual information into
contributions such as synergistic, redundant, or unique information. Despite
PID conceptually being defined for any type of random variables, so far, PID
could only be quantified for the joint mutual information of discrete systems.
Recently, a quantification for PID in continuous settings for two or three
source variables was introduced. Nonetheless, no ansatz has managed to both
quantify PID for more than three variables and cover general measure-theoretic
random variables, such as mixed discrete-continuous, or continuous random
variables yet. In this work we will propose an information quantity, defining
the terms of a PID, which is well-defined for any number or type of source or
target random variable. This proposed quantity is tightly related to a recently
developed local shared information quantity for discrete random variables based
on the idea of shared exclusions. Further, we prove that this newly proposed
information-measure fulfills various desirable properties, such as satisfying a
set of local PID axioms, invariance under invertible transformations,
differentiability with respect to the underlying probability density, and
admitting a target chain rule.
