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Increasing Partitions of Unity

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

Let $(T,\mathcal{B},\mu)$ be a measure and let $f:\bar{\mathbf{R}}\times T \to \bar{\mathbf{R}}$ be a function. Then we say that $f$ is an increasing $\mu$-partition of unity if $f(x,t)$ is increasing in $x$. measurable in $t$ and $\int_Tf(x,t)\mu(dt) =x$ for all $x\in \bar{\mathbf{R}}$. 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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