On Separable Supports of Borel Measures

Some properties of Borel measures with separable supports are considered. In particular, it is proved that any σ-finite Borel measure on a Suslin line has a separable supports and from this fact it is deduced, using the continuum hypothesis, that any Suslin line contains a Luzin subspace with the cardinality of the continuum. Let E be a topological space. We say that the space E has the property (S) if for every σ-finite Borel measure μ defined in this space there exists a separable support, i.e., a separable closed set F (μ) ⊂ E such that μ(E\F (μ)) = 0. Let us consider some examples of topological spaces having the property (S). Example 1. It is obvious that any separable topological space E has the property (S). Example 2. Let E be an arbitrary metric space whose topological weight is not measurable in a wide sense. Then according to the well-known result from the topological measure theory the space E has the property (S). Example 3. Let E be the Alexandrov compactification of some discrete topological space. Then the following statements are equivalent: a) the space E has the property (S); b) card(E) is not measurable in a wide sense. Example 4. Let E be a Hausdorff topological space. We say that E is a Luzin space if every σ-finite diffused (i.e., continuous) Borel measure defined in E is identically zero. The classical Luzin set on the real line R is a Luzin topological space (about Luzin sets see, for example, [1]). One can easily check that any σ-finite Borel measure defined in the Luzin topological space 1991 Mathematics Subject Classification. 54H05.