关于Lebesgue可测性的映射不变性

Mapping invariance of Lebesgue measurability

讨论了在映射Ｔ：Ｒｎ→Ｒｍ下，点集Ｔ（Ａ）Ｒｍ与Ｔ－１（Ｂ）Ｒｎ的可测性，研究了映射Ｔ的保零测性与保可测性的条件．作为推论证明了：当ＥＲｎ可测且ＡＲｎ最多可数时，Ｅ＋Ａ与Ｅ＃Ａ都可测，其中Ｅ＋Ａ＝ｘ＋ａ｜ｘ∈Ｅ，ａ∈Ａ｛｝，Ｅ＃Ａ＝ｘ＃ａ｜ｘ∈Ｅ，ａ∈Ａ｛｝（ｘ＃ａ＝（ｘ１ａ１，ｘ２ａ２，…，ｘｎａｎ））．特别地。

For a mapping T: R n→ R m , the measurability of T(A) and T -1 (B) are discussed, and some conditions, under which the measurability or null measurability of a set is preserved by T , are given. It is proved that if E R n is measurable and A R n is finite or countable, then E+A and E#A are all measurable,where E+A={x+a|x∈E and a∈A } and E#A={(x 1a 1,x 2a 2,…,x na n)|x∈E and a∈A} . Especially, the translation invariance of Lebesgue measure is shown.