Lebesgue density and exceptional points. (English) Zbl 1471.03071
Let \(X\) be a metric space and \(\mu\) a Borel measure on \(X\). We denote by \(B(x;\varepsilon)\) the open ball centred in \(x\) of radius \(\varepsilon\). Let \(A\subseteq X\) be measurable and \(x\in X\). The density of \(x\) is defined by \[\mathscr D_A(x)=\lim_{\varepsilon\to 0}\frac{\mu(A\cap B(x;\varepsilon))}{\mu(B(x;\varepsilon))}.\] Then denote \[\Phi(A)=\{x\in X:\mathscr D_A(x)=1\},\] \[\mathrm{Shrp}(A)=\{x\in X:0<\mathscr D_A(x)<1\},\] \[\mathrm{Blr}(A)=\{x\in X:\mathscr D_A(x)\mbox{ does not exist}\},\] \[\mathrm{Exc}(A)=\mathrm{Blr}(A)\cup\mathrm{Shrp}(A).\] We also denote \(\mathrm{Fr}_\mu(A)\) the set of all \(x\) that, for any open neighborhood \(U\) of \(x\), \(\mu(A\cap U)>0\) and \(\mu(U\setminus A)>0\).
Let \(\mathrm{MEAS}_\mu\) be the \(\sigma\)-algebra of \(\mu\)-measurable sets in \(X\), \(\mathrm{NULL}_\mu=\{A\in\mathrm{MEAS}_\mu:\mu(A)=0\}\), and \[\mathrm{MALG}(X,\mu)=\frac{\mathrm{MEAS}_\mu}{\mathrm{NULL}_\mu}\] endowed with the metric \(\delta([A],[B])=\mu(A\triangle B)\). We also denote \[{\mathscr F}(X,\mu)=\{[C]\in\mathrm{MALG}(X,\mu):C\mbox{ is closed}\},\] \[{\mathscr K}(X,\mu)=\{[K]\in\mathrm{MALG}(X,\mu):K\mbox{ is compact}\}.\] The main results of this article are the following three theorems.
Theorem. Assume that any closed ball of \(X\) is compact, \(\mu(K)<\infty\) for any compact \(K\subseteq X\), and \(\mu(\{x\})=0\) far any \(x\in X\). Then \({\mathscr F}(X,\mu)\) and \({\mathscr K}(X,\mu)\) are \({\mathbf\Pi}^0_3\)-complete in \(\mathrm{MALG}(X,\mu)\).
Theorem. Let \(A\) be a non-empty subset of the Cantor space \(2^\mathbb N\). If \(\mathrm{Int}(A)=\emptyset\) and \(\Phi(A)=A\), then \(\mathrm{Blr}(A)\) and \(\mathrm{Exc}(A)\) are \({\mathbf\Sigma}^0_3\)-complete.
It is worth noting that, by Theoems 1.3 and 1.7 of A. Andretta and R. Camerlo [Adv. Math. 234, 1–42 (2013; Zbl 1264.03096)], the set of \([A]\) with \(\mathrm{Int}(A)=\emptyset\) and \(\Phi(A)=A\) is comeager in \(\mathrm{MALG}(2^\mathbb N,\mu)\), where \(\mu\) is the usual measure on \(2^\mathbb N\).
The authors also present an example of a compact subset \(K\) of \(2^\mathbb N\) such that \(\Phi(K)\) is open and \(\mathrm{Shrp}(K)\) is \({\mathbf\Pi}^0_3\)-complete.
Theorem. Work in \({\mathbb R}^n\) with the \(\ell_p\) metric \((1\le p\le\infty)\) and the Lebesgue measure \(\mu\). For \(A\subseteq{\mathbb R}^n\), if \(A\ne\emptyset,{\mathbb R}^n\) and \(\mathrm{Blr}(A)=\emptyset\), then \(\mathscr D_A(x)=1/2\) for comeager many \(x\in\mathrm{Fr}_\mu(A)\).
The authors also construct examples of \(A\subseteq{\mathbb R}^n\) such that \(\mathrm{Shrp}(A)=\emptyset\) and \(\mathrm{Blr}(A)\ne\emptyset\).
Let \(\mathrm{MEAS}_\mu\) be the \(\sigma\)-algebra of \(\mu\)-measurable sets in \(X\), \(\mathrm{NULL}_\mu=\{A\in\mathrm{MEAS}_\mu:\mu(A)=0\}\), and \[\mathrm{MALG}(X,\mu)=\frac{\mathrm{MEAS}_\mu}{\mathrm{NULL}_\mu}\] endowed with the metric \(\delta([A],[B])=\mu(A\triangle B)\). We also denote \[{\mathscr F}(X,\mu)=\{[C]\in\mathrm{MALG}(X,\mu):C\mbox{ is closed}\},\] \[{\mathscr K}(X,\mu)=\{[K]\in\mathrm{MALG}(X,\mu):K\mbox{ is compact}\}.\] The main results of this article are the following three theorems.
Theorem. Assume that any closed ball of \(X\) is compact, \(\mu(K)<\infty\) for any compact \(K\subseteq X\), and \(\mu(\{x\})=0\) far any \(x\in X\). Then \({\mathscr F}(X,\mu)\) and \({\mathscr K}(X,\mu)\) are \({\mathbf\Pi}^0_3\)-complete in \(\mathrm{MALG}(X,\mu)\).
Theorem. Let \(A\) be a non-empty subset of the Cantor space \(2^\mathbb N\). If \(\mathrm{Int}(A)=\emptyset\) and \(\Phi(A)=A\), then \(\mathrm{Blr}(A)\) and \(\mathrm{Exc}(A)\) are \({\mathbf\Sigma}^0_3\)-complete.
It is worth noting that, by Theoems 1.3 and 1.7 of A. Andretta and R. Camerlo [Adv. Math. 234, 1–42 (2013; Zbl 1264.03096)], the set of \([A]\) with \(\mathrm{Int}(A)=\emptyset\) and \(\Phi(A)=A\) is comeager in \(\mathrm{MALG}(2^\mathbb N,\mu)\), where \(\mu\) is the usual measure on \(2^\mathbb N\).
The authors also present an example of a compact subset \(K\) of \(2^\mathbb N\) such that \(\Phi(K)\) is open and \(\mathrm{Shrp}(K)\) is \({\mathbf\Pi}^0_3\)-complete.
Theorem. Work in \({\mathbb R}^n\) with the \(\ell_p\) metric \((1\le p\le\infty)\) and the Lebesgue measure \(\mu\). For \(A\subseteq{\mathbb R}^n\), if \(A\ne\emptyset,{\mathbb R}^n\) and \(\mathrm{Blr}(A)=\emptyset\), then \(\mathscr D_A(x)=1/2\) for comeager many \(x\in\mathrm{Fr}_\mu(A)\).
The authors also construct examples of \(A\subseteq{\mathbb R}^n\) such that \(\mathrm{Shrp}(A)=\emptyset\) and \(\mathrm{Blr}(A)\ne\emptyset\).
Reviewer: Longyun Ding (Tianjin)
MSC:
| 03E15 | Descriptive set theory |
| 28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
Citations:
Zbl 1264.03096References:
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