Results 1 to 2 of 2

Thread: Sequence in Lebesgue Space

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    22

    Sequence in Lebesgue Space

    Show that if $\displaystyle (f_n)_n$ is a sequence in $\displaystyle L^1[0, 1]$ is such that $\displaystyle \sum_{n=1}^{\infty} || f_n ||_1 < \infty$ then $\displaystyle \sum_{n=1}^{\infty} | f_n(s) | < \infty$ for almost every $\displaystyle s \in [0, 1]$.


    This looks simple to prove. However, I do not see how to prove this right now. Any hints on how to proceed would be appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by pascal4542 View Post
    Show that if $\displaystyle (f_n)_n$ is a sequence in $\displaystyle L^1[0, 1]$ is such that $\displaystyle \sum_{n=1}^{\infty} || f_n ||_1 < \infty$ then $\displaystyle \sum_{n=1}^{\infty} | f_n(s) | < \infty$ for almost every $\displaystyle s \in [0, 1]$.


    This looks simple to prove. However, I do not see how to prove this right now. Any hints on how to proceed would be appreciated.
    You probably know that if $\displaystyle f$ is nonnegative and $\displaystyle \int f<\infty$, then $\displaystyle f(s)<\infty$ for almost every $\displaystyle s$ (this is almost by definition of the integral).

    Then apply that to $\displaystyle f=\sum_{n=1}^\infty |f_n(s)|$... (You will need the monotone convergence theorem to justify the computation of $\displaystyle \int f(=\int \lim_N \sum_{n=1}^N |f_n|$)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lebesgue measure of a manifold embedded in the euclidean space
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Nov 30th 2011, 07:35 PM
  2. Sequence, Lebesgue Space
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Dec 15th 2009, 10:46 AM
  3. Sequence, Lebesgue Space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Dec 13th 2009, 01:53 PM
  4. Question about lebesgue space and fourier transforms
    Posted in the Advanced Math Topics Forum
    Replies: 4
    Last Post: Dec 4th 2008, 10:19 PM
  5. Sequence of Lebesgue measurable functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 14th 2008, 11:51 PM

Search Tags


/mathhelpforum @mathhelpforum