Radon--Nikodym property

A real or complex Banach space $X$ has the Radon--Nikodym property (RNP) if the following holds. For every finite measure space $(\mathrm{\Omega },\mathcal{A},\mu )$ and every norm-countably additive $X$-valued vector measure $\nu$ of bounded variation satisfying $\nu \ll \mu$, there exists a Bochner-integrable function $f:\mathrm{\Omega }\to X$ such that

$\nu (E)={\int }_{E}fd\mu (E\in \mathcal{A})\mathrm{.}$

Such an $f$ is called a Bochner density of $\nu$ with respect to $\mu$. The equality is required on every measurable set, not only on $\mathrm{\Omega }$.

• The control measure $\mu$ is finite; the vector measure separately has bounded variation and is absolutely continuous with respect to it.
• The zero Banach space has RNP, with the zero density for its only vector measure. On the empty measure space the same density satisfies the condition.
• The definition is a property of the Banach target. It is stronger than the scalar Radon--Nikodym theorem when the target is arbitrary.