  We present an operator-free, measure-theoretic approach to the conditional
mean embedding (CME) as a random variable taking values in a reproducing kernel
Hilbert space. While the kernel mean embedding of unconditional distributions
has been defined rigorously, the existing operator-based approach of the
conditional version depends on stringent assumptions that hinder its analysis.
We overcome this limitation via a measure-theoretic treatment of CMEs. We
derive a natural regression interpretation to obtain empirical estimates, and
provide a thorough theoretical analysis thereof, including universal
consistency. As natural by-products, we obtain the conditional analogues of the
maximum mean discrepancy and Hilbert-Schmidt independence criterion, and
demonstrate their behaviour via simulations.
