Towards a Learning Theory of Cause-Effect Inference

The first step towards the deployment of our learning setup is to guarantee the existence of a measure on the space μk(P)⇥L, where μk(P) = {μk(P ) : P 2 P} ✓ Hk is the set of kernel mean embeddings associated with the measures in P . The following lemma provides such guarantee. This allows the analysis within the rest of this Section on μk(P)⇥ L. Lemma 2. Let (Z, ⌧Z) and (L, ⌧L) be two separable topological spaces. Let P be the set of all Borel probability measures on (Z,B(⌧Z)). Let μk(P) = {μk(P ) : P 2 P} ✓ Hk, where μk is the kernel mean embedding (1) associated to some bounded continuous kernel function k : Z ⇥ Z ! R. Then, there exists a measure on μk(P)⇥ L.