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 $Ω\subset {\bf C}^n$. We prove that if $u=v$ on a plurifinely open set $U\subset Ω$ that is Borel measurable, then $(dd^cu)^n|_U=(dd^cv)^n|_U$. 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 $U=\bigcup_{j=1}^m\{φ_j>ψ_j\}$, where $φ_j$, $ψ_j$, $j=1,...,m$, are plurisubharmonic functions on $Ω$.

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