Results 1 to 2 of 2

Math Help - Lebesgue outer measure.

  1. #1
    Junior Member
    Joined
    Nov 2012
    From
    Taiwan
    Posts
    41

    Lebesgue outer measure.

    Let A Rn and let |A|e be the Lebesgue outer measure.
    Suppose A, B Rn and d(A,B)>0. How do I prove that |A B|e=|A|e+|B|e?
    Having thought about this problem for days, I still can't solve it. Any suggestion?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    May 2013
    From
    Czech Republic
    Posts
    28
    Thanks
    1

    Re: Lebesgue outer measure.

    Hi, I would try this:

    If 0<d(A,B)=\inf\{\delta (x,y)\,;\,x\in A\,,\,y\in B\} where \delta (x,y)=\sqrt{\sum_{i=1}^n |x_i-y_i|^2}

    then there exists open set G\subset\mathbb{R}^n such that :
    1. A\subset G
    2. B\cap G=\emptyset .

    Since every open set in (\mathbb{R}^n, \delta) is Lebesgue measurable, there holds

    \forall S\subset\mathbb{R}^n \,:\,|S|_e=|S\cap G|_e + |S - G|_e .

    Rewriting this for S=A\cup B and using properties 1,2 of set G seems to be a good way to solve your problem.
    Last edited by alteraus; November 7th 2013 at 01:43 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Showing that a specific outer measure is a measure
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: August 16th 2011, 11:50 AM
  2. [SOLVED] Outer measure as Metric
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 8th 2010, 07:54 AM
  3. 2 simple outer measure problems
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: September 18th 2010, 06:55 AM
  4. Outer measure and measurable sets.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 3rd 2009, 06:57 AM
  5. Outer Measure
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: July 24th 2009, 06:44 AM

Search Tags


/mathhelpforum @mathhelpforum