Building the system 为了满足测度在极限运算下的可列可加要 求,需要回答? 如何寻求一个比环R更大的集类L,其中每个 元素上都可分配一个正实数测度
Building the system • 为了满足测度在极限运算下的可列可加要 求，需要回答？ • 如何寻求一个比环R更大的集类L,其中每个 元素上都可分配一个正实数测度？
A system of sets A C P(Q)is called a o-algebra if 1.0,9∈A 2.fA∈4, then ca∈A 3.fAn∈.4(forn∈N), then Un An∈4 因此我们需要一个对无穷次并或交运算封闭对sgma环或代数。 H(P)={EECx,存在B∈R(=1,2,3,…)使EC∪E} i=1 按定义,这是个sgma环, so it should be ok as our new workspace How to define measure on h(R)? Possibly use m on R?回忆如何定义圆面积
因此我们需要一个对无穷次并或交运算封闭对sigma环或代数。 按定义，这是个sigma环， so it should be ok as our new workspace How to define measure on H(R)? Possibly use m on R? 回忆如何定义圆面积
Outer measure 定义231如果是环R上的测度,在H(R)上作集函数μ*:当E∈ H(R)时 ()=imf∑(EE1∈REC∪E 讠=1 μ*称为由测度μ所引出的外测度 Is it consistent with measure defined on r? yes, if E belongs to R, then ustar is a measure since E can be covered by itself Why use infinitely number of elements in R? For measurable set, we can cover it perfectly with infinitely number of sets but may not do so with finite number of sets Is it a measure?-next slides
Outer measure • Is it consistent with measure defined on R? - yes, if E belongs to R, then ustar is a measure since E can be covered by itself. • Why use infinitely number of elements in R? – For measurable set, we can cover it perfectly with infinitely number of sets but may not do so with finite number of sets. • Is it a measure? - next slides
Key Property of Outer Measure 定理231设是由环R上测度μ所引出的外测度,那么对于任何一 列E;∈H(R),成立不等式 ∪B≤∑(E (232) In other words u star is not a measure on H(R)-subadditivity < countable additivity So, what we will do?
Key Property of Outer Measure • In other words, u_star is not a measure on H(R) – subadditivity <> countable additivity • So, what we will do?
Seeking R We want to find a subset of H(R),r, such that 1. onr the out measure u* is a measure 2. R* should be a sigma ring -it's the domain of a measure 3. R* should cover R, why? It's the extension of R: we actually use the measure on R to get the measure onr some consistent should be preserved
Seeking R* • We want to find a subset of H(R), R*, such that – 1. on R*, the out measure u* is a measure. – 2. R* should be a sigma ring – it’s the domain of a measure. – 3. R* should cover R, why? It’s the extension of R: we actually use the measure on R to get the measure on R*, some consistent should be preserved