可测空间原子的一个应用

An Application of Atoms in Measurable Spaces

设(Ω,F)是一个可测空间,f:Ω→S是一个映射.周知,若f满足双射条件,则f(F)构成S上的一个σ-代数.本文利用(Ω,F)的原子获得了一个比双射条件严格弱的新条件,在此新条件下f(F)仍然构成S上的一个σ-代数,此外还利用所获定理给出了一个已知结果的非常简洁的证明.

Let （Ω,F） be a measurable space and f：Ω →S a map to S. It is well known that f（F） forms a sigma - algebra over S if f satisfies the bijective condition. In this note, by using the atoms in （Ω ,F）, the authors found a weaker condition under which f（F） forms a sigma - algebra over S . In addition, as an application of the main result, a very simple proof of a known theorem was presented.