Proposition 12.10.6.B62E802ED631 Let $0\leq p\in \mathcal D_\omega $ be nonzero and put $R=\rho _\omega (p)$. Suppose $T_ j\in \mathscr G_\omega $ converge weakly to $T$ and satisfy $D_ p(T_ j)\to R$. If the densities $p k_{T_ j}$ are uniformly integrable, then
\[ D_ p(T)=R,\qquad p k_{T_ j}\, \mathrm{d}V\rightharpoonup p k_ T\, \mathrm{d}V. \]
Consequently, a maximizing sequence for $D_ p$ with uniformly integrable densities $pk_{T_ j}$ has a maximizing current among its weak cluster limits.
Proof.
For positive integers $K$, put $p_ K=\min (p,K)$. Since $p$ is finite almost everywhere, uniform integrability gives
\[ \varepsilon _ K:=\sup _ j\int _{\{ p>K\} }p k_{T_ j}\, \mathrm{d}V \longrightarrow 0. \]
For every nonnegative continuous function $h$, the weight $hp_ K$ is bounded and measurable, so the upper semicontinuity of $D_{hp_ K}$ proved in theorem 12.10.1 gives
\[ \limsup _ j\int _ Xhp k_{T_ j}\, \mathrm{d}V \leq \int _ Xhp_ K k_ T\, \mathrm{d}V +\| h\| _\infty \varepsilon _ K. \]
Now let $K\to \infty $. For $h=1$ this yields $R\leq D_ p(T)$, and the supremum formula (12.10.2) gives the reverse inequality. If $\Lambda $ is a weak cluster limit of the measures $p k_{T_ j}\, \mathrm{d}V$, the same estimate for all nonnegative continuous $h$ gives $\Lambda \leq p k_ T\, \mathrm{d}V$. Both measures have mass $R$, so they agree. Since these measures have bounded masses, they form a weakly relatively compact family, and every cluster limit has now been identified. The final assertion follows from the compactness of $\mathscr G_\omega $.
$\square$
Pome
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