Corollary 5.34.3C6F4FBF73B9E

Proof. In a coordinate chart put $\kappa =\mathrm{dd}^{\mathrm{c}}|z|^2$ and $\mu =T\wedge \kappa ^{n-j}$. By the Lelong monotonicity formula in bidimension $(n-j,n-j)$, the function $r\mapsto r^{-2(n-j)}\mu (B(x,r))$ is nondecreasing; see [Chapter III, formulas (5.5)–(5.8), pp. 158–159, DemCADG]. Fix a relatively compact inner chart and a radius $r_0>0$ such that all balls of radius $r_0$ centered in it lie in a fixed compact subset of the chart. Their masses are bounded by the mass of that compact set. Hence

uniformly for centers in the inner chart. A Borel set in the inner chart with zero Hausdorff $2(n-j)$-measure can be covered by balls $B(x_ i,r_ i)$ with $\sum _ ir_ i^{2(n-j)}$ arbitrarily small, so it has zero $\mu $-mass. Since $T$ is positive, the total variation of each of its coefficient measures is bounded by a constant multiple of $\mu $. A countable cover by charts proves the assertion. $\square$