用Loeb测度构造正则及τ－光滑Borel测度

Construction of regular and τ smooth Borel measures by Loeb measures

设（Ｘ，Ｔ）是Ｔ２空间，Ｔ１是拓扑Ｔ的基，Ａ是＊Ｘ上的内代数，ν是Ａ上的内测度，本文用无穷小方法证明了：在ｋ－饱和的非标准模型中，若Ｘ是正则空间，且对每一Ｔ∈Ｔ１，＊Ｔ∈Ａ，则对Ｘ的每一Ｂｏｒｅｌ集Ｂ，ｓｔ－１（Ｂ）∈Ｌｕ（Ａ）∩ｎｓ（＊Ｘ），且νｓｔ－１｜Ｆ（Ｘ）是τ－光滑Ｂｏｒｅｌ测度，νｓｔ－１｜Ｆ（Ｘ）是Ｒａｄｏｎ测度；若对每一Ｔ∈Ｔ，＊Ｔ∈Ａ，并且对每一Ｔ∈Ｔ及每一ε∈Ｒ＋，有闭集ＣＴ，使Ｌ（ν）（＊Ｔ－＊Ｃ）＜ε，则对每一Ｂｏｒｅｌ集Ｂ，ｓｔ－１（Ｂ）∈Ｌ（ν，Ａ）∩ｎｓ（＊Ｘ）且νｓｔ－１｜Ｆ（Ｘ）是正则τ－光滑Ｂｏｒｅｌ测度．

Let ( X ,T) be a T 2 space, T 1 be a base of T, A be an internal algebra over * X, ν be an internal measure on A. In k saturated nonstandard model, the following results are proved by using infinitesimal method: If X is regular, and for each T ∈T 1, * T ∈A, then for every Borel subset B of X , st -1 (B) ∈L u(A)∩ns( * X ), and ν st -1 |F( X ) is τ smooth Borel measure, ν st -1 |F( X ) is Radon measure; If for each T ∈ T , * T ∈T,and for arbitrary T ∈T and arbitrary ε∈R +, there exists a closed set C T,such that L(ν )( * T- *C) <ε, then for every Borel subset B of X , st -1 ( B )∈ L(ν ,A)∩ns( * X ),and ν  st -1 |F(X) is a regular and τ smooth Borel measure. Furthermore, the applications of these results in the extension of measures are discussed.