## Kolodziej’s acta paper “the complex monge-ampere equation”——a detailed ploblem [closed]

Recently,I am reading kolodziej's acta paper,there are some ditails that i do not know clearly.

In the top line of page 99,"it's no restriction to assume that for each s we have $\nu(\cup_{I\in{B_s}}\partial I)=0$",i do not know why.Here $\nu=(dd^{c}v)^n$ and $v\in PSH(\Omega)\cap L^{\infty}$,$\partial I$ is the boundary of some cube.

From Kolodziej's view,a poriori $\nu(\cup_{I\in{B_s}}\partial I)$ may not be zero.However,i think $\partial I$ can be seen as a part of a pluripolar set,then according to Bedford and Taylor's reasult,we know the above monge-ampere measure concentrates no mass on $\partial I$.So we get $\nu(\cup_{I\in{B_s}}\partial I)=0$,and it should not be a assumption!

I hope some expert in this field can help me.Thanks.

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## closed as too localized by Franz Lemmermeyer, Angelo, Andres Caicedo, Daniel Moskovich, Ryan BudneyDec 29 2010 at 6:26

One way to argue is this: We have $\nu(\Omega)<1$. (For example a $\sigma$-finite Borel measure would suffice.) If you fix the subdivisions $\beta_s$ of $I_0$, then for $\mathcal{H}^{2n}$-almost every $t \in C^n$ we have $\nu(\partial (I+t))=0$ for every $I \in \bigcup_s \beta_s$. Therefore you can move the grid slightly to have the claim satisfied.