匿名使用者
匿名使用者 發問時間: 科學數學 · 8 年前

question of real analysis

Suppose E is subset of IR^2 ,E={(x,y): x=y , 0<=x<=1,0<=y<=1},is E have measure zero?

1 個解答

評分
  • Sam
    Lv 6
    8 年前
    最佳解答

    In the Lebesgue measure on R^2,the measure is zero, since we can find a series {cov(n) : n=1,2,…} of coverings of E such that their measures converge to zero. The open rectangle Cov(n) is boubded by thefollowing four lines :y-x=1/n, y-x=-1/n, x+y=-1/n and x+y=1+1/n.Obvious, the area or measure of the open rectangle cov(n)=(sqrt(2)/n) *(1+sqrt(2)/n) -->0as n-->0.

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