An equality of Monge-Ampère measures

Let $u$ and $v$ be two plurisubharmonic functions in the domain of definition of the Monge-Ampère operator on a domain $\mathrm{\Omega }\subset {\mathbf{C}}^n$. We prove that if $u=v$ on a plurifinely open set $U\subset \mathrm{\Omega }$ that is Borel measurable, then $(d{d}^{c}u{)}^n{\mathrm{\mid }}_U=(d{d}^{c}v{)}^n{\mathrm{\mid }}_U$. This result was proved by Bedford and Taylor in the case where $u$ and $v$ are locally bounded, and by El Kadiri and Wiegerinck when $u$ and $v$ are finite, and by Hai and Hiep when $U$ is of the form $U={\bigcup }_{j=1}^m\{{\phi }_j>{\psi }_j\}$, where ${\phi }_j$, ${\psi }_j$, $j=1,\mathrm{.}\mathrm{.}\mathrm{.},m$, are plurisubharmonic functions on $\mathrm{\Omega }$.