Cover and Thomas provides definition of Conditional Mutual Information (CMI) for discrete random variables but doesn't say anything about continuous variables. Wikipedia has a section about a "more general definition" of CMI that is applicable to continuous case, which looks a bit involved given my current knowledge.
My question is, can't we extend the definition of CMI from discrete case to continuous case as follows :
$$I(X;Y \mid Z) = \iiint f_{X,Y,Z}(x,y,z)\log \frac{f_z(z)\ \ f_{X,Y,Z}(x,y,z)}{f_{X,Z}(x,z)\ \ f_{Y,Z}(y,z)} \, dx \, dy \, dz$$
where the $\ f$ s are density functions.
Is this definition wrong ? If yes, what is wrong here conceptually and what is the simplest correct definition of CMI for continuous variables ?