Results 1 to 2 of 2

Math Help - Lebesgue measurable function and Borel measurable function

  1. #1
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374

    Lebesgue measurable function and Borel measurable function

    Let f(x) be a real-valued Lebesgue measurable function on \mathbb R^n, show that there exist Borel measurable function g and h such that g(x)=h(x) a.e.[m], and g(x)\leq f(x)\leq h(x)\text{ for all }x\in \mathbb R^n.
    "a.e.[m]" means " holds almost everywhere with respect to lebesgue measure".

    I know this problem is good problem ofr helping us understand the relation between lebesgue and borel measurability. It concerns the completion of Borel measurable set class. And something else.....
    Any help will be appreciate.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    this problem is difficult. I am open to any idea or solution.
    My idea:
    lebesgue measurable set can be written as the Union of a borel measurable set and a set with measure 0.
    and the Lemma 9 and proposition 10 in Page 260-261 may help us. (see Royden's Real Analysis.)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. a question about Lebesgue measurable function
    Posted in the Differential Geometry Forum
    Replies: 15
    Last Post: June 29th 2011, 09:53 PM
  2. Borel measurable
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: September 30th 2010, 05:34 AM
  3. Borel measurable function
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 11th 2010, 03:21 PM
  4. lebesgue measurable function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 22nd 2009, 08:11 AM
  5. Replies: 2
    Last Post: October 1st 2009, 07:07 PM

Search Tags


/mathhelpforum @mathhelpforum