線上人數: 169

10602 高等微積分二

第17講 8.2 Volume and sets of measure zero

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L17A


L17_A
        8.2 Volume and sets of measure zero
              (1) Defintion: Let A⊂ R^n be a bounded set 
              (2) Note
              (3) Property: All rectangles B =[a,b]x...x[an,bn] ⊂ R^n
                                  has volume = ∏ (bi-ai) 
              (4) Question:
                 
 1.怎樣的 bounded set 會有體積呢?
                       2.A has volume zero <=> 1A is integreable and ∫1A = 0
 
 
L17_B
        8.2 Volume and sets of measure zero
              (1)Question: A has volume zero <=>
                                  1A is integreable and ∫1A = 0
              (2)Property: A bounded set A in R^n has zero
              (3)Cor: 1.If A has volume zero, then any subset of
                              A has volume zero
                  1.Any finite union of volume zero sets also has volume zero
 
 
L17_C
        8.2 Volume and sets of measure zero
              (1) Property: A bounded set A in R^n has zero
              (2) Definition: A set A ⊂ R^n
                                   (not necessary bounded) has measure zero,...
              (3) Property: 1.Any subset of measure zero set is also measure zero
                   2.A set in R^n having volume is not measure zero.
                   3.If A has volume zero, then A is measure zero
                   4.Any single point set in R^n has volume zero and measure zero
                   5.The real line, regarded as a subset of R^2 has measure zero,
                          but as a subset of R,it does not. 
 
              (4) Thm: Suppose that the sets A1,A2,...have
                             measure zero in R^n. Then the union UAi
                             has measure zero
              (5) Cor:Any countable set in R^n in measrue zero
              (6) Property: measure zero ≠>volume zero
 

L17_D
        8.3 Lebesque's thm
              (1) Question: 衡量函數值
              (2) Definition
              (3) Thm (Lebesque's thm)
              (4) Remark

 

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