关于在闭集上连续的函数在多面体上的一个连续逼近
On the Approximation of a Function Continuous off a Closed Set by One Continuous Off a Polyhedron
摘要：
设$P$是一个有限单纯复形，其下空间为（$P$中单纯形的并集）$|P|$。设$Q$是$P$的一个子复形。让$a \geq 0$。然后存在具有以下属性的$K < \infty$、\emph{depending only on $a$ and $Q$,}。设$\mathcal{S} \subset |P|$是闭的，且$Φ$是$|P| \setminus \mathcal{S}$到某个拓扑空间$\mathcal{F}$的连续映射.假设$\dim (\tilde{\mathcal{S}} \cap |Q|) \leq a$，其中“$\dim$“= Hausdorff维数。则存在$\tilde{\mathcal{S}} \subset |P|$使得$\tilde{\mathcal{S}} \cap |Q|$是$Q$的子复形的底层空间，并且存在$|P| \setminus \tilde{\mathcal{S}}$到$\mathcal{F}$的连续映射$\tildeΦ$使得$\mathcal{H}^{a} \bigl(\tilde{\mathcal{S}} \cap |Q| \bigr) \leq K \mathcal{H}^{a} \bigl(\mathcal{S} \cap |Q| \bigr)$，其中$\mathcal{H}^{a}$表示$a$维Hausdorff测度，若$x \in \tilde{\mathcal{S}}$则$x$属于$P$中与$\mathcal{S}$相交的单形;如果$x \in |P| \setminus \mathcal{S}$、$x \in σ\in P$和$σ$不与$Q$中的任何单纯形相交，其单纯内部与$\mathcal{S}$相交，则定义$\tildeΦ(x)$并等于$= Φ(x)$ ;如果$σ\in P$，则$\tildeΦ(σ\setminus \tilde{\mathcal{S}}) \subset Φ(σ\setminus \mathcal{S})$ ;若$\mathcal{F}$是度量空间，且$Φ$是$|P| \setminus \mathcal{S}$上的局部Lipschitz，则$\tildeΦ$是$|P| \setminus \tilde{\mathcal{S}}$上的局部Lipschitz。
英文摘要：
Let $P$ be a finite simplicial comple with underlying space (union of simplices in $P$) $|P|$. Let $Q$ be a subcomplex of $P$. Let $a \geq 0$. Then there exists $K < \infty$, \emph{depending only on $a$ and $Q$,} with the following property. Let $\mathcal{S} \subset |P|$ be closed and suppose $Φ$ is a continuous map of $|P| \setminus \mathcal{S}$ into some topological space $\mathcal{F}$. Suppose $\dim (\tilde{\mathcal{S}} \cap |Q|) \leq a$, where "$\dim$" = Hausdorff dimension. Then there exists $\tilde{\mathcal{S}} \subset |P|$ such that $\tilde{\mathcal{S}} \cap |Q|$ is the underlying space of a subcomplex of $Q$ and there is a continuous map $\tildeΦ$ of $|P| \setminus \tilde{\mathcal{S}}$ into $\mathcal{F}$ such that $\mathcal{H}^{a} \bigl(\tilde{\mathcal{S}} \cap |Q| \bigr) \leq K \mathcal{H}^{a} \bigl(\mathcal{S} \cap |Q| \bigr)$, where $\mathcal{H}^{a}$ denotes $a$-dimensional Hausdorff measure; if $x \in \tilde{\mathcal{S}}$ then $x$ belongs to a simplex in $P$ intersecting $\mathcal{S}$; if $x \in |P| \setminus \mathcal{S}$, $x \in σ\in P$, and $σ$ does not intersect any simplex in $Q$ whose simplicial interior intersects $\mathcal{S}$, then $\tildeΦ(x)$ is defined and equals $= Φ(x)$; if $σ\in P$ then $\tildeΦ(σ\setminus \tilde{\mathcal{S}}) \subset Φ(σ\setminus \mathcal{S})$; and if $\mathcal{F}$ is a metric space and $Φ$ is locally Lipschitz on $|P| \setminus \mathcal{S}$ then $\tildeΦ$ is locally Lipschitz on $|P| \setminus \tilde{\mathcal{S}}$ Moreover, $P$ can be replaced by an arbitrarily fine subdivision without changing $K$.
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