2
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$\{f_n\}$ is a sequence of continuous functions with following properties:

  1. $0 \leq f_n(x) \leq 1, \forall x \in \Bbb R$
  2. $f_n(x)$ is monotonically decreasing sequnce as $n\to \infty$
  3. The Limiting function $f$ is not Riemann integrable.

I know that such function is Lebesgue integrable using Monotone convergence theorem but not sure about Riemann integrable.

If there is counterexample please give me that. Or give me a hint so that I can prove this theorem

Any help will be appreciated.

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  • $\begingroup$ I think you can take the counter-example from here math.stackexchange.com/questions/612098/… $\endgroup$ – Yanko Mar 10 at 13:52
  • $\begingroup$ Dear Sir, That $f_n$ is not continuous also not monotonically decreasing $\endgroup$ – SRJ Mar 10 at 14:00
  • $\begingroup$ Oh yeah I didn't see you want $f_n$ to be continuous. (You could work out the monotonically decreasing by taking $1-f_n$). $\endgroup$ – Yanko Mar 10 at 14:01
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    $\begingroup$ Now I think I'm not wrong that the answer is in the comment here math.stackexchange.com/questions/2670955/… (open the wikipedia page, 4th bullet) $\endgroup$ – Yanko Mar 10 at 14:09
  • $\begingroup$ By 2 do you mean $f_1(x) \ge f_2(x) \ge \cdots $ for each $x?$ $\endgroup$ – zhw. Mar 10 at 17:47

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