Lemma 5.28.1200AB47766404

Proof. Take a nonnegative smooth radial mollifier $\chi _\delta $ supported in the ball of radius $\delta $. Then $T*\chi _\delta \to T_{\mathrm{ac}}$ almost everywhere. Indeed, the convolutions of the absolutely continuous parts converge at Lebesgue points of their densities, while the convolution of the singular part $\lambda _ s$ of a coefficient measure is bounded by $C\delta ^{-2n}|\lambda _ s|(B_\delta )$, which tends to zero almost everywhere by the differentiation theorem for measures. Applying lemma 5.28.10 with $g=F^{1/n}$ and letting $\delta \downarrow 0$ gives $\det _\beta T_{\mathrm{ac}}\geq F$.

For the reverse inequality, write the convolved current as the Hessian of a decreasing regularization of a locally bounded plurisubharmonic function plus a convergent smooth correction. The continuity of bounded products gives $(T*\chi _\delta )^ n\rightharpoonup T^ n$. Fatou's lemma, applied against a nonnegative compactly supported test function, then gives $\det _\beta T_{\mathrm{ac}}\, \beta ^ n\leq T^ n=F\beta ^ n$. $\square$