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By "not-too-pathological" I intended at least to include the requirement "measurable".

I just notice that I used the wrong metric. The article uses the Hausdorf metric and I used the measure of the symetric difference.

And the article assume that the sets are compact, so they are measurable as you say. Anyway compact sets can be quite pathological (but not as pathological as non measurable sets).

It also suggests the argument generalizes to symbols as non-compact sets.

I made a comment elsewhere on this thread that explains that symbols themselves being compact isn't so important, but that the set of descriptions of the symbols must be compact. For example, if the description of the symbol is not the symbol itself as a set, but a map f:[0,1]^2 -> [0,1] that describes the "intensity" of ink at each point, then the natural conclusion is that the description of a symbol must be upper semicontinuous, which makes the set of descriptions compact.

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