Pome

Lemma 19.6.2322B2EF8281A7

Lemma 19.6.23.22B2EF8281A7 Let $(\Delta ^ i)_{i\geq 1}$ be an increasing sequence in $\mathcal{K}_ n$ with Hausdorff limit $\Delta $ such that $\operatorname{vol}\Delta ^ i>0$ for all $i\geq 1$. Consider an increasing sequence $(\Delta ^ i_{\bullet })_{i\geq 1}$ with $\Delta ^ i_{\bullet }\in \mathrm{TC}(\Delta ^ i)$ for all $i\geq 1$. Let $\Delta _{\bullet }\in \mathrm{TC}(\Delta )$ be defined as

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    $\Delta _{\max }=\lim _{i\to \infty }\Delta ^ i_{\max }$;

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    for any $\tau <\Delta _{\max }$, $\Delta _{\tau }$ is the Hausdorff limit of $\Delta ^ i_{\tau }$.

Then we have

\[ \operatorname{DH}(\Delta _{\bullet }^ i)\rightharpoonup \operatorname{DH}(\Delta _{\bullet }). \]

Proof. Let $f\in C_ c(\mathbb {R})$. It follows from proposition 19.6.16 that

\[ G[\Delta _{\bullet }^ i]\to G[\Delta _{\bullet }] \]

almost everywhere on $\Delta $. Extending $f(G[\Delta _{\bullet }^ i])$ by $0$ outside $\Delta ^ i$, we also have $\mathds {1}_{\Delta ^ i}\to \mathds {1}_{\Delta }$ almost everywhere on $\Delta $, since the boundary of the convex body $\Delta $ has Lebesgue measure zero. The dominated convergence theorem gives

\[ \int _{\Delta ^ i} f(G[\Delta _{\bullet }^ i])\, \mathrm{d}\lambda \to \int _{\Delta } f(G[\Delta _{\bullet }])\, \mathrm{d}\lambda . \]

This is the weak convergence. $\square$

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