Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral ?

  1. Oct 18, 2012 #1
    "All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral"?

    I am trying to learn about the Lebesgue integral and Lebesgue measurability. None of my text books really cover it from the basics, but I found this document online which seems to be pretty through in explaining the motivation behind developing the Lebesgue integral http://web.media.mit.edu/~lifton/snippets/measure_theory.pdf [Broken]


    However, there is a statement I am having a hard time grasping, on the bottom of the first page:

    "Third, all Lp spaces except for L fail to be complete under the Riemann
    integral"

    Here is what I understand: when saying "Lp spaces" I'm assuming this means metric spaces, right? I know from functional analysis that a complete metric space is one where there are no "gaps", or formally, where every Cauchy sequence has a limit that's also in the space. (That's why, for example the rational numbers with the st. Euclidean metric (Lp with p=2) is not complete, because we have gaps at all the irrational places)


    Here's what I don't understand: what does it mean to be complete under the Riemann integral? I don't understand what this means. I thought a metric space would be a set of numbers, with a metric defined on it, and it would be complete or incomplete just based on that information alone. Where does the Reimann integral come in to play?
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Oct 18, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: "All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral

    No, the [itex]L^p[/itex]-spaces are a special kind of Banach space. They are defined as (for [itex]1\leq p <+\infty[/itex]):

    [tex]L^p([a,b])=\left\{f:[a,b]\rightarrow \mathbb{R} ~\text{measurable}~\left|~ \int_a^b |f|^p < +\infty \right.\right\}[/tex]

    The norm on this space is

    [tex]\|f\|_p=\sqrt[p]{\int_a^b |f|^p}[/tex]

    The thing is that the above integral must be the Lebesgue integral. If we just focus on the Riemann-integral, then we find out there are not enough Riemann integrable functions and the space will be incomplete.
     
  4. Oct 18, 2012 #3
    Re: "All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral

    oh ok, thank you very much for clearing up my confusion
     
  5. Oct 21, 2012 #4

    Bacle2

    User Avatar
    Science Advisor

    Re: "All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral

    An example for L^1(ℝ) , which you can generalize:

    Take an enumeration {an} of the Rationals, and Let XS be

    the characteristic function of the set S .

    Define:

    f_1=χa1

    ...
    fna1+.......xan

    .....

    So that your function fn is 1 at each of the rationals ≤ an, and is

    0 everywhere else. Then each of the fn is R-integrable, but the

    sequence converges to the characteristic function of the Rationals, which is not

    R-integrable.

    Notice the limit functions is Lebesgue integrable. As a nitpick, remember: Riemann, not Reimann; I don't mind so much, but your

    prof. may cringe.
     
    Last edited: Oct 21, 2012
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral ?
  1. Completely regular spaces (Replies: 125)

Loading...