Theorem 23.1.747C4D0FB8D775

Proof. Put $m=b-6$. Necessity follows from lemma 23.1.72. Finite convex combinations of the measures $mh_ p$ are shrinking defects by lemma 23.1.73, proposition 23.1.70, the latter applied to an atomic measure $\rho $ supported on unitary maps that move a fixed fiber to the desired fibers. Approximate an arbitrary base measure of mass $m$ by finite atomic measures of the same mass, using finite partitions of diameter tending to zero. Weak continuity of $\mathcal L$ and the diagonal choice on dense test functions used in the preceding proof give a realizing sequence for every prescribed sequence $\varepsilon _ j$.

Every nonempty open subset of a fiber has positive Haar measure. Local trivializations therefore show that $\mathcal L\mu $ has support $\pi ^{-1}(\operatorname {supp}\mu )$, and the fiber measures have no point atoms. If $B$ is a pluripolar subset of the base, local plurisubharmonic functions whose pole sets contain $B$ pull back by the submersion to such functions on $X$. Hence $\pi ^{-1}(B)$ is pluripolar, and $\mathcal L\mu (\pi ^{-1}(B))=\mu (B)$; thus $\mathcal L\mu $ charges a pluripolar set whenever $\mu $ does.

Conversely, suppose that $\mu $ charges no pluripolar subset of the base. Cover a pluripolar set in $X$ by countably many local product charts, each carrying a plurisubharmonic function $q(\xi ,w)\not\equiv -\infty $ whose pole set contains the part of the set in that chart. For a relatively compact disk $D$ in the fiber direction, the function

is plurisubharmonic and not identically $-\infty $ on a smaller base chart. Indeed, local integrability of $q$ and Fubini's theorem show that $F$ is locally integrable, and in particular finite almost everywhere; a common upper bound and the reverse Fatou lemma give upper semicontinuity; and integrating the submean inequalities in the base directions gives the submean property. The restriction of $q$ to a fiber is either subharmonic with a pole set of zero area or identically $-\infty $. The fibers of the second kind lie over the pluripolar set $\{ F=-\infty \} $, which has zero $\mu $-measure. Integrating the Haar measures of the fibers against $\mu $ therefore shows that each local pole set has zero $\mathcal L\mu $-measure, and the countable cover proves the assertion for this pluripolar set. For arbitrary pluripolar subsets, the same argument applies to Borel pole sets containing them, with outer measure. Similarly, for the first implication, a Borel $\mathcal L\mu $-null set containing $\pi ^{-1}(B)$ has a measurable fiber-mass function equal to one on $B$, which gives the corresponding outer-measure statement. $\square$