Results 1 to 2 of 2

Math Help - Is a bounded integrable function square integrable?

  1. #1
    Junior Member
    Joined
    Oct 2009
    From
    London
    Posts
    42

    Is a bounded integrable function square integrable?

    If f \in L^1(E) is bounded, and E is of finite measure, i.e m(E)<\infty, (m here is the lebesgue measure but it can be arbitrary).

    Is f \in L^2(E)?

    Is it enough to say:
    since |f|<M for some real M.
    \int _E \left|f\right|^2dm<\int _EM^2dm=M^2\int _Edm=M^2m(E)<\infty

    But what I dont get is that this argument only uses the fact f is bounded, not the fact that f is integrable.

    I know that L^2(E) \subseteq L^1(E) since E is of finite measure. I wonder if this has something to do with the answer?
    Last edited by aukie; May 28th 2010 at 06:04 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Oct 2009
    From
    London
    Posts
    42
    Perhaps if i write |f|^2=|f||f|<M|f| then

    \int _E \left|f\right|^2dm<\int _EM|f|dm=M\int _E|f|dm<\infty

    then it would be true that if f is integrable then it is also square integrable... does this seem plausible?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Example of function not integrable and the square is
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: July 6th 2011, 02:45 AM
  2. Function Bounded and Continuous on (0,1) but not Integrable!
    Posted in the Differential Geometry Forum
    Replies: 13
    Last Post: May 15th 2010, 07:34 AM
  3. bounded and continuous function not integrable
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 11th 2010, 11:16 PM
  4. Real Analysis: Prove BV function bounded and integrable
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 17th 2010, 02:31 PM
  5. Replies: 0
    Last Post: December 1st 2008, 06:43 PM

Search Tags


/mathhelpforum @mathhelpforum