The distance problem via subadditivity

In a recent paper, Aldous, Blanc and Curien asked which distributions can be expressed as the distance between two independent random variables on some separable measured metric space. We show that every nonnegative discrete distribution whose support contains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>0</mml:mn> </mml:math> arises in this way, as well as a class of compactly supported distributions with density.