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arXiv · 2309.11868

A Radon-Nikodym theorem for monotone measures

Yao Ouyang·Jun Li
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Abstract

A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone measures is proved. Without any presumptive condition, we obtain a necessary and sufficient condition for the ordered pair $(μ, ν)$ of finite monotone measures to have the so-called Radon-Nikodym property related to a nonnegative measurable function $f$. If $ν$ is null-continuous and weakly null-additive, then $f$ is uniquely determined almost everywhere by $ν$ and thus is called the Radon-Nikodym derivative of $μ$ w.r.t. $ν$. For $σ$-finite monotone measures, a Radon-Nikodym type theorem is also obtained under the assumption that the monotone measures are lower continuous and null-additive.

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Yao Ouyang, Jun Li. 2023-09-21. A Radon-Nikodym theorem for monotone measures. https://arxiv.org/abs/2309.11868

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