Loeb空间中的Radon-Nikodym导数(英文)

Radon Nikodym Derivative on Loeb Space

英文关键词:absolute continuity  Loeb space  Radon-Nikodym Theorem  Radon-Nikodym derivative.

中文摘要:

      设υ及μ为定义在可测空间（Ｘ，Ｓ）上的有限测度．本文首先证明了若υ《μ（即υ关于μ绝对连续），则有Ｌ（^*Ｓ，^*μ）(?)Ｌ（＊Ｓ，υ）．进而证明了υ《μ当且仅当Ｌ（^*υ）《Ｌ（^*μ）并且ｄ（Ｌ（^*υ））／ｄ（Ｌ（^*μ））＝^０（^*（ｄυ／ｄμ））即Ｌｏｅｂ空间中的Ｒａｄｏｎ－Ｎｉｋｏｄｙｍ定理．本文按一种自然的方式定义了σ－有限测度空

英文摘要:

      In this paper, we first show that if 《υ is absolutly continuous with respect to μ , i.e., υ《u , then L( ^*S, ^*μ)(?)L( ^*S, ^*υ) . We also prove that υ《μ if and only if L( ^*υ)《L(^*μ) and d(L(^*υ))/d(L(^*μ))= ⁰( ^*(dμ/dυ)) . We shall define the Loeb space of σ-finite measure space by a natural way and prove that the results above can be extended to σ-finite measure spaces.