Proposition 5.14.5E10B5F00D3ED

Proof. Put $u=\sup_i\varphi_i$ and $v=u^*$. By the pluripolar discrepancy theorem, $\{u<v\}$ is pluripolar. The stronger local conclusion in the upper-envelope theorem says that the same discrepancy set is real polar in each coordinate chart. Thus both asserted notions of polarity follow from the proved envelope construction, rather than from an assumed measurability of an arbitrary supremum. The conclusion is local, as are the definitions of polar and pluripolar subsets of $X$.