  This paper extends the proof of density of neural networks in the space of
continuous (or even measurable) functions on Euclidean spaces to functions on
compact sets of probability measures. By doing so the work parallels a more
then a decade old results on mean-map embedding of probability measures in
reproducing kernel Hilbert spaces. The work has wide practical consequences for
multi-instance learning, where it theoretically justifies some recently
proposed constructions. The result is then extended to Cartesian products,
yielding universal approximation theorem for tree-structured domains, which
naturally occur in data-exchange formats like JSON, XML, YAML, AVRO, and
ProtoBuffer. This has important practical implications, as it enables to
automatically create an architecture of neural networks for processing
structured data (AutoML paradigms), as demonstrated by an accompanied library
for JSON format.
