Paper:
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.

Review:
The authors study in this paper the approximation capabilities of neural networks for real valued functions on probability measure spaces (and on tree structured domains). 
The first step of the paper consists in extending standard NN results to probability measure spaces, that is rather than having finite dimensional vectors as inputs, the NN considered here have probability measures as inputs. The extension to this case is straightforward and closely related to older extension on infinite dimensional spaces (see for instance the seminal paper of Stinchcombe https://doi.org/10.1016/S0893-6080(98)00108-7 and e.g. http://dx.doi.org/10.1016/j.neunet.2004.07.001 for an application to NN with functional inputs). Nothing quite new here.
In addition, and exactly as in the case of functional inputs, the real world neural networks do not implement what is covered by the theorem but only an approximation of it. This is acknowledged by the authors at the end of Section 2 but in a way that is close to hand waving. Indeed while the probability distribution point is valuable and gives interesting tools in the MIL context, the truth is that we have no reason to assume the bag sizes will grow to infinite or even will be under our control. In fact there are many situations were the bag sizes are part of the data (for instance when a text is embedded in a vector space word by word and then represented as a bag of vectors). Thus proving some form of universal approximation in the multiple instance learning context would need to take this fact into account, something that is not done at all here. 
Therefore I believe the contribution of this paper to be somewhat limited.