Results 1 to 3 of 3

Thread: Need to prove |f(x)|<= 1 almost every where.

  1. #1
    Member
    Joined
    Aug 2009
    Posts
    78

    Need to prove |f(x)|<= 1 almost every where.

    Let $\displaystyle f :X \rightarrow R$ be a measurable function in $\displaystyle X$ with $\displaystyle m(X)< \infty$.
    Show that if $\displaystyle f^n$ is Lebesgue integrable for every $\displaystyle n$ and that $\displaystyle \lim_{n\to\infty}\int f^n\,dm$ exists in $\displaystyle R$,then $\displaystyle -1\le f(x)\le 1$ almost every where.

    Can anyone help me with this problem??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Nov 2006
    From
    Florida
    Posts
    228
    Here is just a general idea. Assume that $\displaystyle |f|>1$ on some measurable set with $\displaystyle m(U)>0$. Then $\displaystyle f^n>M$ for any $\displaystyle M\in\mathbb{R}$ once $\displaystyle n$ is large enough. So what will this do to $\displaystyle \int f^ndm$?

    Hint: $\displaystyle \int f^ndm=\int_U f^ndm+\int_{U^c}f^ndm$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Just to elaborate a bit on putnam120's suggestion, for k=1,2,3,..., let $\displaystyle S_k = \{x\in X: |f(x)|\geqslant 1+1/k\}$. Then $\displaystyle \int_X\!|f^n|\,dm\geqslant \int_{S_k}\!\!|f^n|\,dm\geqslant (1+1/k)^nm(S_k)$. If $\displaystyle m(S_k)\ne0$ then this goes to infinity as $\displaystyle n\to\infty$. Also, if n is even then $\displaystyle |f^n| = f^n$, so $\displaystyle \lim_{n\to\infty}\int_X\!f^n\,dm$ will not exist in that case.

    Therefore $\displaystyle m(S_k) = 0$ for all k and hence $\displaystyle \textstyle m\left(\bigcup_{k}S_k\right) = 0$. But $\displaystyle \textstyle \bigcup_{k}S_k = \{x\in X:|f(x)>1\}$. Therefore $\displaystyle -1\leqslant f(x)\leqslant 1$ almost everywhere.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove a/b and a/c then a/ (3b-7c)
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Mar 23rd 2010, 05:20 PM
  2. prove,,,
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Mar 1st 2010, 09:02 AM
  3. Prove |w + z| <= |w| +|z|
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Feb 28th 2010, 05:44 AM
  4. Replies: 2
    Last Post: Aug 28th 2009, 02:59 AM
  5. How to prove that n^2 + n + 2 is even??
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Nov 30th 2008, 01:24 PM

Search Tags


/mathhelpforum @mathhelpforum