Results 1 to 2 of 2

Math Help - Counterexample for convergence in L1

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    Counterexample for convergence in L1

    Suppose that  \{ f_n \} \subset L^1( \mu ) and  f_n converges uniformly to f [/tex], by a theorem that I know, f \in L^1 ( \mu ) and  \int f_n \rightarrow \int f provided that  \mu (X) < \infty , X being the domain set.

    Now, if  \mu (X) = \infty , this may not be true.

    Counterexample:

    Define f_n(x)= \left \{ \begin {array} {cc} \frac {1}{x} & \mbox { if }<br />
1 \leq x < n \\ 0 & \mbox { if } x \geq n \end {array} \right .

    And define  f(x)= \left \{ \begin {array} {cc} \frac {1}{x} & \mbox { if }<br />
1 \leq x \\ 0 & \mbox { if } x \leq 1 \end {array} \right .

    And f_n \rightarrow f uniformly, but f_n is integrable for all n while f is not.

    Question:

    Q1: Why is f_n \rightarrow f uniformly? The book just claim it is, but I'm trying to prove it using the defintion of  \forall \epsilon > 0 \exists N \in \mathbb {N} such that  \mid f_n-f \mid < \epsilon

    Q2: I understand why f is not integrable since it goes to infinity, but are f_n integrable? Is it because  \int f_n = \int _1^n \frac {1}{x} is finite?

    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    1) I assume your domain is \mathbb{R} notice that to evaluate \vert f_n(x)-f(x) \vert we need three cases:

    -) If x< 1 then f_n(x)=f(x)=0
    -) If 1\leq x \leq n then f_n(x)=f(x)=\frac{1}{x}
    -) If n<x then \frac{1}{x}  \leq \frac{1}{n} and \vert f_n(x) -f(x) \vert = \frac{1}{x} < \epsilon for n big enough.

    2) Notice that f_n are Riemann integrable.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Counterexample Help
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: October 5th 2010, 08:24 AM
  2. Counterexample to uniformly convergence
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: December 3rd 2009, 07:59 AM
  3. Counterexample
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: October 28th 2009, 03:06 PM
  4. counterexample of convergence of integral again
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: May 1st 2009, 06:36 PM
  5. A counterexample
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: November 1st 2006, 09:06 AM

Search Tags


/mathhelpforum @mathhelpforum