Results 1 to 2 of 2

Math Help - Lebesgue integral

  1. #1
    Junior Member
    Joined
    Mar 2010
    From
    Melbourne
    Posts
    28
    Thanks
    1

    Lebesgue integral

    I'm trying to evaluate

     \int_{-\infty}^{\infty}\frac{\lambda}{\vert x \vert}

    where \lambda is the Lebesgue measure on \mathbb{R} and the following is what I have done:

    Let f_{n}(x)=\frac{1}{\vert x \vert}\chi_{[-n,n]}. Then since (f_{n}:n\in\mathbb{N}) is non-negative and increasing, by the Monotone Convergnece Thorem we have

    \int_{-\infty}^{\infty}\frac{\lambda}{\vert x \vert}= \int_{\mathbb{R}}\lim_{n}\frac{1}{\vert x \vert}\chi_{[-n,n]}d\lambda = \lim_{n}\int_{\mathbb{R}}\frac{1}{\vert x \vert}\chi_{[-n,n]}\\=\lim_{n}\Big(\int_{(-n,0)}-\frac{1}{x}\lambda+\int_{(0,n)}\frac{1}{x}\lambda \Big)\hspace{1mm}  =\infty,

    since the integrals inside the limit notation coincides with Riemann integral.

    Am I on the right track?
    Last edited by willy0625; June 7th 2010 at 02:30 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Feb 2010
    From
    Berlin
    Posts
    16
    You're on the right track.
    Jerry
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lebesgue Integral
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 12th 2010, 03:02 PM
  2. Lebesgue integral
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: January 9th 2010, 05:01 PM
  3. Lebesgue integral
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 23rd 2009, 12:12 AM
  4. Lebesgue Integral
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: December 22nd 2009, 05:35 PM
  5. Lebesgue Integral
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 9th 2009, 01:47 AM

Search Tags


/mathhelpforum @mathhelpforum