asked in 科學數學 · 1 decade ago


Let I be a generalized rectangle in R^n and let the function f:I→R be integrable.Denote the interior of I by D. Show that the restriction f:D→R

is integrable and that ∫_I▒f(下限是I)=∫_D▒f (下限是D)



1 Answer

  • 1 decade ago
    Favorite Answer

    Claim: If D has Jordan content 0, f is integrable on D then ∫Df(x)dx=0

    Proof of claim: Choose a compact rectangle A⊃D Define g(x)=f(x) if x∈D

    g(x)=0 if x∈A\D

    It suffices to show that ∫Ag(x)dx=0

    f is bounded function let M=sup{|f(x)| :x∈D}=sup{|g(x)|:x∈A}=M

    Suppose M>0

    Since D has Jordan content 0, for ε>0 there is partition P of A ,P={P1 P2,…,PM} so that 0≤U(χD P)<ε/M

    Then for any partition P’ finer than P,we have


    So g is integrable and has value 0

    Now if I is open rectangle then I=D and ∫Df(x)dx=∫If(x)dx

    If not then I=∂I∪D and ∂I∩D=∅

    ∂I has Jordan content 0 =>∫∂If(x)dx=0




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