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On computability and disintegration

Published online by Cambridge University Press:  28 July 2016

NATHANAEL L. ACKERMAN ,
CAMERON E. FREER  and
DANIEL M. ROY
NATHANAEL L. ACKERMAN
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A. Email: nate@math.harvard.edu
CAMERON E. FREER
Affiliation:
Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, and Gamalon Labs, Cambridge, Massachusetts, U.S.A. Email: freer@mit.edu
DANIEL M. ROY
Affiliation:
Department of Statistical Sciences, University of Toronto, Toronto, Ontario, Canada Email: droy@utstat.toronto.edu
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Abstract

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We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim. When a measure does not have a unique continuous disintegration, we may still obtain a disintegration when some basis of continuity sets has the Vitali covering property with respect to the measure; the disintegration, however, may depend on the choice of sets. We show that, when the basis is computable, the resulting disintegration is strongly Weihrauch reducible to Lim, and further exhibit a single distribution realizing this upper bound.

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