Results 1 to 2 of 2

Math Help - Lebesgue measure problem

  1. #1
    Newbie
    Joined
    Sep 2011
    Posts
    3

    Lebesgue measure problem

    Hi,

    I have the following problem:

    Let m be the Lebesgue measure on R and let v be a measure on the Lebesgue sigma-algebra such that:
    (i) v is absolutely cont. with respect to m.
    (ii) v({x}+A)=v(A) for all x in R and A in the Lebesgue sigma-algebra.
    Show: v=c*m for some constant c in R.

    Really all I can see is that (and this is given by assumption (i)) the statement holds for all E such that m(E)=0. Other then that, nada. Any hints?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: Lebesgue measure problem

    Apply Radon-Nikodym theorem: it will give you an integrable function f such that for all A Lebesgue-measurable, \nu(A)=\int_A f(t)dm(t). Now, fix x\in\mathbb R, and put h(u):=f(u+x)-f(u). h is integrable and for all A Lebesgue-measurable, \int_A h(t)dm(t)=0. It shows that f is constant.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lebesgue measure on RxR
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: August 17th 2011, 06:24 AM
  2. Lebesgue measure > +ve
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 10th 2010, 01:06 PM
  3. lebesgue measure
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 11th 2010, 02:42 AM
  4. lebesgue measure
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 27th 2009, 05:35 AM
  5. Lebesgue measure of some spaces...
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: April 17th 2009, 10:46 AM

Search Tags


/mathhelpforum @mathhelpforum