Results 1 to 2 of 2

Math Help - F is integrable

  1. #1
    Member
    Joined
    Jan 2010
    Posts
    88

    F is integrable

    Suppose f is integrable on [a,b], show |f| is integrable on [a,b];
    I know I have to show Sp|f|-sp|f|<=Spf-spf
    But I'm stuck from there
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,386
    Thanks
    1476
    Awards
    1
    Quote Originally Posted by calculuskid1 View Post
    Suppose f is integrable on [a,b], show |f| is integrable on [a,b];
    I know I have to show Sp|f|-sp|f|<=Spf-spf
    But I'm stuck from there
    The usual way goes something like this.
    Define two new functions: F^ +  (x) = \left\{{\begin{array}{rl}<br />
   {F(x),} & {0 \leqslant F(x)}  \\   {0,} & {F(x) < 0}  \\ \end{array} } \right.\;\& \,F^ -  (x) = \left\{ {\begin{array}{r,f}   { - F(x),} & {F(x) < 0}  \\   {0,} & {F(x) \geqslant 0}  \\ \end{array} } \right.

    We see that |F(x)|=F^+(x)+F^-(x) so show each of those is integrable.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. f & g Riemann integrable, show fg is integrable
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: February 12th 2011, 09:19 PM
  2. f is integrable
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: January 31st 2011, 01:31 PM
  3. Is a bounded integrable function square integrable?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 28th 2010, 06:26 PM
  4. Prove that if f is integrable then kf is integrable.
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 9th 2010, 12:35 PM
  5. [SOLVED] f integrable implies f^2 integrable
    Posted in the Differential Geometry Forum
    Replies: 15
    Last Post: June 8th 2009, 10:53 PM

Search Tags


/mathhelpforum @mathhelpforum