A description/proof of that open set complement of measure 0 subset is dense
Topics
About: metric space
About: measure
About: map
The table of contents of this article
Starting Context
- The reader knows a definition of metric space.
- The reader knows a definition of measure.
- The reader knows a definition of denseness.
Target Context
- The reader will have a description and a proof of the proposition that for any open set on any metric space, the complement of any measure 0 subset with respect to the open set is dense on the open set.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any metric space, M, any open set, \(U \subseteq M\), and any measure 0 subset, \(S \subseteq U\), the complement, \(S^c = U \setminus S\), of \(S\) with respect to \(U\) is dense on \(U\).
2: Proof
Centered at any point, \(p \in U\), there is any small-enough open ball, \(B_{p-\epsilon} \subseteq U\). \(S^c \cap B_{p-\epsilon} = (U \setminus S) \cap B_{p-\epsilon} = U \cap B_{p-\epsilon} \setminus S \cap B_{p-\epsilon}\). \(m (S^c \cap B_{p-\epsilon}) = m (U \cap B_{p-\epsilon}) - m (S \cap B_{p-\epsilon}) = m (B_{p-\epsilon}) - 0\). As \(m (B_{p-\epsilon}) \gt 0\), \(m (S^c \cap B_{p-\epsilon}) > 0\), which means that \(S^c \cap B_{p-\epsilon}\) is not empty, which means that for any point, \(p \in U\), and any small-enough open ball, \(B_{p-\epsilon} \subseteq U\), there is a point of \(S^c\) inside the open ball, which means that \(S^c\) is dense.
