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Disintegration of Gaussian measures and average-case optimal algorithms. (English) Zbl 1131.41010

The paper deals with the problem of the existence of a disintegration of a given probability measure relative to a measurable mapping and its use for the study of average-case optimal algorithms. Precisely, the authors study disintegration of measures in Banach spaces with respect to continuous linear operators. The main result is the following. Let \(X\), \(Y\) be separable Banach spaces, \(B(X)\) the Borel \(\sigma\)-algebra of \(X\), \(\mu\) a Gaussian measure in \(X\) and \(\eta: X\to Y\) a continuous linear operator. Then there exists a disintegration \(q: B(X)\times Y\to [0,1]\) of \(\mu\) with respect to \(\eta\) and, moreover, the measures \(q(\cdot, y)\), \(y\in Y\) are Gaussian. The authors also show that, by using this result, it is possible to prove the existence and to find an explicit form of average-case optimal algorithms.

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47A50 Equations and inequalities involving linear operators, with vector unknowns
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