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The sequence of $\alpha$-trimmings of empirical probabilities is shown toconverge, in the Painlev\'{e}--Kuratowski sense, on the class of probabilitymeasures endowed with the weak topology, to the $\alpha$-trimming of thepopulation probability. Such a result is applied to the study of the asymptoticbehaviour of central regions based on the trimming of a probability.

Consistency of the $\alpha$-trimming of a probability. Applications to central regions