An example of pointwise non-convergence of iterated conditional expectation operators |
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Authors: | Mustafa Akcoglu Jonathan L King |
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Institution: | (1) University of Toronto, Toronto, Canada;(2) University of Florida, 32611-2082 Gainesville, FL, USA |
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Abstract: | Given ∈, we construct a sequence
, … of Borel sub-sigma-algebras on the unit interval with the following property. Suppose the identity functionf(x)=x is transformed by successive conditioning on
, then
, then
, Then the lim sup, with respect ton, will exceed (pointwise almost-everywhere) 1−∈ and its lim inf will be less than ∈.
The sequence of functions also will fail to converge in the
. This contrasts with the long-open conjecture that if all the
come from a finite set of sigma-algebras, then the resulting sequence of functions must converge in
.
J. L. King was partially supported by NSF grant DMS-9112595. |
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Keywords: | |
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