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We provide a theoretical foundation for non-parametric estimation offunctions of random variables using kernel mean embeddings. We show that forany continuous function $f$, consistent estimators of the mean embedding of arandom variable $X$ lead to consistent estimators of the mean embedding of$f(X)$. For Mat\'ern kernels and sufficiently smooth functions we also providerates of convergence. Our results extend to functions of multiple randomvariables. If the variables are dependent, we require an estimator of the meanembedding of their joint distribution as a starting point; if they areindependent, it is sufficient to have separate estimators of the meanembeddings of their marginal distributions. In either case, our results coverboth mean embeddings based on i.i.d. samples as well as "reduced set"expansions in terms of dependent expansion points. The latter serves as ajustification for using such expansions to limit memory resources when applyingthe approach as a basis for probabilistic programming.

Consistent Kernel Mean Estimation for Functions of Random Variables