[分析] 測度 與 metric

[ericakk (愛瑞卡)]

Let (Χ,Μ,μ) is a finite measure space,

1. Say that E～F, if μ(EΔF) = 0 ; then ～ is an equivalence relation on Μ.

2.For E,F in Μ, define ρ(E,F) = μ(EΔF),

then ρ(E,G)≦ρ(E,F) + ρ(F,G),

hence, ρ defines a metric on the space Μ/～ of equivalence classes.

這個證明 part 1，我已經check ～ is an equivalence relation on Μ.如下：

a. reflexive: if μ(EΔE) = 0, then μ(E) = μ(E)

b. symmetric: if μ(EΔF) = 0 ,then μ(E) = μ(F)

c. transitive: if μ(EΔF) = 0 and μ(EΔG) = 0 , then μ(E) = μ(F) = μ(G)



至於 part 2 的  metric 的前兩個性質也很容易證得：

a.ρ(E,E) = μ(EΔE) = 0

b.ρ(E,F) = μ(EΔF) = μ(FΔE) = ρ(F,E)

c.但我不會這個 ρ(E,G)≦ρ(E,F) + ρ(F,G)

 該怎麼寫呢？感謝... 
※ 編輯: ericakk         來自: 140.119.149.234      (11/01 11:17)
推 ppia        :EΔFㄈ(EΔG)u(FΔG),畫個文氏圖(Venn diagram)
→ ppia        :就看得出來了
→ ericakk     :謝謝你^.^