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Aarhus Universitets segl

Increasing Partitions of Unity

by J. Hoffmann-Jørgensen
Research Reports Number 502 (February 2009)

Let (T,B,μ) be a measure and let f:ˉR×TˉR be a function. Then we say that f is an increasing μ-partition of unity if f(x,t) is increasing in x. measurable in t and Tf(x,t)μ(dt)=x for all xˉR. Increasing partitions of unity have a variety of applications which will be explored in the paper. For instance, applications include the Fubuni-Tonelli theorem for upper and lower integrals and Fubuni-integrals, measurability or upper (lower) semicontinuity of integral transforms, and construction of functions with a prescribed integral transform and satisfying a given set of (in)equalities.

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