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Suppose that a sequence of integrable function $(f_n)$ converges uniformly to the zero function. I want to find an example of such $f_n$ exhibiting the property that $$\int f_n \not \to 0.$$

What if we replace uniform convergence with uniformly bounded and point wise convergence?

1 Answers1

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Does $f_n=\frac{1}{n}\chi_{[0,n]}$ work? The integral of all these are $1$, but they converge uniformly to the $0$ function.

If you want a compact domain (or even just a finite measure domain), and your functions $f_i$ are uniformly bounded and pointwise convergent, then you use Dominated Convergence theorem on the constant function that bounds them all.

Also, I don't know why your bounty answer needs to draw from a credible or official source. If you believe and understand the answer, why does it matter where it comes from?

Bob Jones
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  • What if I want a finite measure domain, point wise convergence of $f_n \to 0$, but not necessarily uniformly bounded? –  Feb 19 '17 at 04:15
  • Counterexample: take a bump function on $[0, 1]$, and then dilate it by $\frac{1}{2}$ in the $x$ direction and $2$ in the $y$ direction, and repeat. – Bob Jones Feb 19 '17 at 04:18
  • @user3359 On $(0,1),$ let $f_n(x) = nx^n.$ Then $f_n\to 0$ pointwise on $(0,1),$ yet $\int_0^1 f_n = n/(n+1) \to 1.$ – zhw. Feb 19 '17 at 22:39
  • @BobJones Can you give me an example of where the convergence is point wise, not uniform, but the functions are uniformly bounded? I'll then be happy to award you the bounty. –  Feb 22 '17 at 23:17
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    Try a bump function on $[0, 1]$ and then translate it to the right by $1$ repeatedly. – Bob Jones Feb 23 '17 at 06:55
  • @BobJones What will the integral of the bump function converge to? –  Feb 26 '17 at 04:03
  • Whatever the integral of the original bump function is. It's constant. – Bob Jones Feb 26 '17 at 08:42
  • @user3359 Why did I not receive the full bounty? Did you forget to accept, or were my answers not to your satisfaction? – Bob Jones Feb 27 '17 at 05:28