Lemma 19.6.22.9C05354365F4 Let $(\Delta ^ i)_{i\geq 1}$ be a decreasing sequence in $\mathcal{K}_ n$ with limit $\Delta $. Suppose that $(\Delta _{\bullet }^ i)_{i\geq 1}$ is a decreasing sequence with $\Delta _{\bullet }^ i\in \mathrm{TC}(\Delta ^ i)$. Suppose that there is $\Delta _{\bullet }\in \mathrm{TC}(\Delta )$ such that
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$\Delta _{\max }=\lim _{i\to \infty }\Delta ^ i_{\max }$;
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for any $\tau <\Delta _{\max }$, we have $\Delta ^ i_{\tau }\xrightarrow {d_{\mathrm{Haus}}}\Delta _{\tau }$.
Then $\operatorname{DH}(\Delta _{\bullet }^ i)\rightharpoonup \operatorname{DH}(\Delta _{\bullet })$.
Proof.
Let $f\in C_ c(\mathbb {R})$. It follows from proposition 19.6.14 that
\[ G[\Delta _{\bullet }^ i]\to G[\Delta _{\bullet }] \]
pointwise on $\Delta $. Hence the dominated convergence theorem gives
\[ \int _{\Delta } f(G[\Delta _{\bullet }^ i])\, \mathrm{d}\lambda \to \int _{\Delta } f(G[\Delta _{\bullet }])\, \mathrm{d}\lambda . \]
Since $\Delta ^ i\xrightarrow {d_{\mathrm{Haus}}}\Delta $, theorem 19.7.5 gives $\operatorname{vol}(\Delta ^ i\setminus \Delta )\to 0$. The contribution of $\Delta ^ i\setminus \Delta $ is bounded by $\| f\| _{\infty }\operatorname{vol}(\Delta ^ i\setminus \Delta )$, so
\[ \int _{\Delta ^ i} f(G[\Delta _{\bullet }^ i])\, \mathrm{d}\lambda \to \int _{\Delta } f(G[\Delta _{\bullet }])\, \mathrm{d}\lambda . \]
This is the weak convergence.
$\square$
Pome
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