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.