PomeProposition 18.8.4.B919F0D14EE2 Finite intersections of the relations $d_ O<r$, $d_ I<r$, $d_\pi <r$ and $\mathcal U(t,a)$, with $r,t>0$ and $a>1$, form a base of entourages for a uniformity on $\mathfrak S(X,\omega )$. Denote its topology by $\tau _{\omega ,0}$, where the subscript $0$ records the zero obstacle in the envelopes. A net $[u_ j]$ converges to $[u]$ if and only if both envelopes converge in $L^1$, $d_\pi ([u_ j],[u])\to 0$ for every fixed $\pi $, and $([u_ j],[u])\in \mathcal U(t,a)$ eventually for every fixed $t,a$. Smooth positive choices of $\mathrm dV$ and of the $\gamma _\pi $ do not change the uniformity.
Proposition 18.8.4B919F0D14EE2
Proof. All generators are symmetric and contain the diagonal. Multiplier ideals decrease as the weight increases, as one sees from local integrability after subtracting a local upper bound from the potential. For $b=\sqrt a$, put $\mathcal V=\mathcal U(t,b)\cap \mathcal U(bt,b)$. If $([x],[y]),([y],[z])\in \mathcal V$, then
\[ \mathcal I(atx)\subseteq \mathcal I(bty)\subseteq \mathcal I(tz), \]
and the same argument with $x$ and $z$ exchanged shows that $\mathcal V\circ \mathcal V\subseteq \mathcal U(t,a)$. For the pseudometric relations, halving the radius gives the corresponding refinement under composition. Taking finite intersections of these refinements proves the axioms of a uniformity, and convergence in it means eventual membership in each generating entourage. Bounded ratios of smooth positive volume forms and lemma 18.8.1 show that the auxiliary choices do not matter. $\square$
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