View : 28 Download: 0

ON THE EXTENSION OF MEASURE

Title
ON THE EXTENSION OF MEASURE
Authors
高舜惠
Issue Date
1977
Department/Major
대학원 의학과
Publisher
梨花女子大學校 大學院
Degree
Master
Advisors
趙泰根
Abstract
本稿에서는 algebra 위에 定義된 測度(measure)로 부터 誘導된 外測度(outer measure)를 이용한 測度의 擴張에 관하여 硏究하였다. Algebra 위에 定義된 測度로부터 그 algebra로 生成된 σ-algebra 위로 擴張된 測度는 complete도 saturated도 아니란 것을 보이는 것이 本稿의 중요 內容이다. 또한 測度로 부터 外測度를 構成하는 여러가지 방법을 集約 比較하였다. 첫째 : γ(Φ) = 0 그리고 γ(G) ≥ 0 이라는 條件을 만족시키는 擴大實價函數(extended real valued function) γ가 sequential covring class □위에 定義되어 있을때 임의의 集合 E에 대하여 μ*(E) = inf□γ(G_n)으로 μ*(E)를 定義하면 μ*는 外測度가 된다. 外測度라는 것은 x의 멱集合 P(X) 위에 定義된 擴大實價函數로서 다음 條件을 만족하는 것을 의미한다 : 01. μ*(Φ) = 0. 02. A⊂B⇒μ*(A)≤μ*(B). 03. E⊂U □ E_(i) ⇒μ*(E)≤ □ μ*(E_(i)). 둘째 : 測度 μ가 ring R 위에 定義되어 있을때 ring R로 生成된 hereditary σ-ring H(R)의 임의의 원소 E에 대하여 μ*(E) = inf {∑□ μ(E_(n)) 1 E_(n)∈R,n=1,2,3 ?? , E⊂U □E_(n) }이라 定義하면 μ*는 H(R)위에서의 外測度이다. 셋째 : μ가 algebra □ 위에 定義된 測度일때 임의의 集合 E에 대하여 μ*(E) = inf {∑□ μ(A_(i)) 1 A_(i)∈□,i = 1,2,3 ?? , E⊂U □A_(i) }라고 定義하면 μ*는 X의 멱集合上의 外測度가 된다. 이 μ*를 測度 μ로부터 誘導된 外測度라고 한다. μ가 algebra□위에 定義된 測度이고 μ*가 μ로부터 誘導된 外測度이면 μ*를 μ* - 可測集合族( the family of μ* - measurabl sets)上으로 縮小시킨 □는 μ의 擴張이다. 이때 (1) μ* -可測集合들로 이루어진 σ-algebra □ 上의 測度 □ 는 μ의 擴張으로 complete고 satruated다. 그러나 (2) □를 algebra □로 生成된 σ-algebra σ(□)에 縮小시킨 □는 μ의 擴張이지마는 complete로 satruated도 아니다. 라는 사실이 本論文의 要旨이다.;In this paper, we shall study the extension theory of measure using the outer measure induced by a measure on an algebra. The main obJect of this paper is to show that the extension measure from a measure on an algebra is neither complete nor saturated on the smallest σ-algebya generated by the algebra. We also compare several methods of constructing outer measures out of measures defined on a variety of classes of sets, We briefly describe the methods of constructing outer measures: Method 1. An extended real valued set function λ on a sequential covering class □ is defined as a function such that λ(Ø) = 0 and λ(G) ≥ 0 for each G in □ and for eaeh set E⊂X we define □^*(E) = inf Σ_(n)λ(G_n) where the infimum is taken as {G_n} varies over all possible □-covering of E, then □^* is an outer measure. An outer measure means an extended real valued set function defined on P(X), the power set of X, with the following properties: 01. □^*(Ø) = 0, 02. If A⊂B then□^*(A)≤□^*(B), 03. If E⊂U^∞_i=l E_i then □^*(E) ≤Σ^∞_i=l □^*(E_i). Method 2. If □ is a measure on a ring λ, then for every E in H(λ) we define □^*(E) = inf{Σ^∞_i=l □(E_i) : E_i∈λ, I=l,2,…, E⊂U^∞_i=l E_i}, then □^* is an outer measure on H(λ) where H(λ) is the hereditary σ-ring generated byλl. Method 3. Let □ be a measure on an algebra□. For each set E⊂X, we define □^*(E) = inf Σ^∞_i=l □(A_i) where {A_i} ranges over all sequences from □ such that E⊂U^∞_i=lA_i. Then □^* becomes an outer measure. This □^* is called an. outer measure induced by □, Let □ be a measure on an algebra □ and □^* be an outer measure induced by □, then the restriction □ of □^* to the □^*-measurable sets is an extension of □. In the last section we have shown the following main facts of this paper: (1) The extension □ of a measure □ on an algebra to the σ-algebra □ which is the class of all □^*-measurable sets is complete and saturated [the Proposition in III]. (2). The exbension □ of □ to the smalleat σ-a1gebra σ(□) containing □ is neither complete nor saturated [Example 1 and Example 2 In III].
Fulltext
Show the fulltext
Appears in Collections:
일반대학원 > 의학과 > Theses_Master
Files in This Item:
There are no files associated with this item.
Export
RIS (EndNote)
XLS (Excel)
XML


qrcode

Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.

BROWSE