My Math Forum Give an example of a sequence of integrable functions that..

 Real Analysis Real Analysis Math Forum

 February 9th, 2009, 03:04 PM #1 Newbie   Joined: Feb 2009 Posts: 2 Thanks: 0 Give an example of a sequence of integrable functions that.. Can you guys think of any exampleS? Give an example of a sequence of integrable functions {fn} (going from 1 to infinity) on [0,1] that converges pointwise to a non integrable function f:[0,1]-->R
 February 9th, 2009, 05:15 PM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Give an example of a sequence of integrable functions that.. Riemann or Lebesgue integrable? If it's Riemann, consider $f_n:[0,1]\mapsto\mathbb{R}$ defined by $f_n(x)=\begin{cases} 1\text{ if }x\in\mathbb{Q},\ x=p/q\text{ in lowest terms and }q\leq n,\\ 0\text{ otherwise.} \end{cases}$ Then, $\int_0^1 f_n(x)\,dx= 0,$ but as $n\mapsto\infty,\ \ f_n$ converges pointwise to the function $\chi(x)=\begin{cases}1\text{ if }x\in\mathbb{Q},\\ 0\text{ if }x\notin\mathbb{Q},\end{cases}$ which is not Riemann integrable (but is Lebesgue integrable). If you want a series of functions that converge pointwise to a non-Lebesgue-integrable function, then it's too late for my brain to work sufficiently...
 February 9th, 2009, 05:32 PM #3 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Give an example of a sequence of integrable functions that.. OK, maybe it's not too late... consider $f_n:[0,1]\mapsto\mathbb{R}$ given by $f_n(x)=\begin{cases}0,\quad &x=0,\\x^{\frac1n-1},\quad &x=>0.\end{cases}=$ Then $\int_0^1f_n(x)\,dx=n,$ and $f_n$ converges pointwise to the function $f(x)=\begin{cases}0,\quad&x=0,\\x^{-1},\quad&x=>0,\end{cases}=$ which is not integrable on [0,1].

 Tags functions, give, integrable, sequence

,

### sequence of integrable functions

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post kapital Calculus 10 July 26th, 2012 08:36 AM veronicak5678 Real Analysis 1 April 15th, 2012 02:12 PM guynamedluis Real Analysis 8 September 30th, 2011 09:37 AM sharp Elementary Math 3 October 27th, 2010 03:36 PM eskimo343 Complex Analysis 0 January 27th, 2010 06:42 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top