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We show that the incompatibility of a set of measurements cannot be increasedby subjecting them to a filter, namely, by combining them with a device thatpost-selects the incoming states on a fixed outcome of a stochastictransformation. This result holds for several measures of incompatibility, suchas those based on robustness and convex weight. Expanding these ideas toEinstein-Podolsky-Rosen steering experiments, we are able to solve the problemof the maximum steerability obtained with respect to the most general localfilters in a way that allows for an explicit calculation of the filteroperation. Moreover, our results generalize to nonphysical maps, i.e., positivebut not completely positive linear maps.

Measurement incompatibility cannot be stochastically distilled