PomeProposition 13.15.15.ACA3FA133A3B For a family of nonpositive potentials in $\mathrm{PSH}(X,\omega )$, the uniform tail condition of theorem 13.15.13 is equivalent to a uniform bound on $J_{\chi ,\omega }$ for some continuous nondecreasing unbounded function $\chi \geq 1$.
Proposition 13.15.15ACA3FA133A3B
Proof. One implication was proved in corollary 13.15.14. Conversely, write the family as $(u_ i)_{i\in I}$. As in the proof of theorem 13.15.13, the tail condition and lemma 13.15.3 give a uniform bound for the total masses. Choose numbers $T_ k\to +\infty $, with $T_{k+1}>T_ k+1$, such that $\mu _{u_ i}(\{ -u_ i>T_ k\} )\leq 2^{-k}$ for every $i\in I$. Let $\rho _ k$ be a continuous nondecreasing function, equal to zero on $[0,T_ k]$ and to one on $[T_ k+1,+\infty )$, and define
\[ \chi (s)=1+\sum _{k\geq 1}\rho _ k(s). \]
The sum is locally finite, so $\chi $ is continuous, nondecreasing and unbounded. Tonelli's theorem gives
\[ J_{\chi ,\omega }(u_ i) \leq \mu _{u_ i}(X)+\sum _{k\geq 1}\mu _{u_ i}(\{ -u_ i>T_ k\} ) \leq \sup _{i\in I}\mu _{u_ i}(X)+1, \]
so this weight gives a uniform integrability bound for the whole family. $\square$
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