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)
has volume = ∏ (bi-ai)
(4) Question:
1.怎樣的 bounded set 會有體積呢?
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
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
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,...
(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
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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