2

$\{f_n\}$ 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.

Andrews
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Curious student
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  • I think you can take the counter-example from here https://math.stackexchange.com/questions/612098/an-example-of-a-sequence-of-riemann-integrable-functions-f-n-that-converges?rq=1 – Yanko Mar 10 '19 at 13:52
  • Dear Sir, That $f_n$ is not continuous also not monotonically decreasing – Curious student Mar 10 '19 at 14:00
  • Oh yeah I didn't see you want $f_n$ to be continuous. (You could work out the monotonically decreasing by taking $1-f_n$). – Yanko Mar 10 '19 at 14:01
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    Now I think I'm not wrong that the answer is in the comment here https://math.stackexchange.com/questions/2670955/pointwise-limit-of-continuous-functions-but-not-riemann-integrable (open the wikipedia page, 4th bullet) – Yanko Mar 10 '19 at 14:09
  • By 2 do you mean $f_1(x) \ge f_2(x) \ge \cdots $ for each $x?$ – zhw. Mar 10 '19 at 17:47
  • @zhw. Yes Sir ... – Curious student Mar 10 '19 at 23:57

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