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Let $f:[a,b]\to\mathbb{R}$ be a bounded function. I want to prove that if there exist sequences of tagged partitions $(P_n)_{n=1}^{\infty}$ and $(Q_n)_{n=1}^{\infty}$ of the interval $[a,b]$, so that:

$$\inf_{n\geq 1} U(f,P_n)=\sup_{n\geq 1}L(f,Q_n)=I$$

($U$ is the upper sum and $L$ the lower sum), then f is integrable and $\int_{a}^{b}f(x)\; dx= I$

I've tried and I've got to nothing really...

carmichael561
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nono
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    are you sure bounded is sufficient? I mean... If you take the characteristic function of the rationals I am quite sure you will not reach the result you claim you want to prove. See this question http://math.stackexchange.com/questions/1079172/is-the-indicator-function-of-the-rationals-riemann-integrable – b00n heT Jun 07 '16 at 19:17
  • In that case (Dirichlet function) all lower sums are $0$ and all upper sums are $1$, so there are no partition sequences with the property specified in OP. This is not a valid counterexample. – RRL Jun 07 '16 at 20:10
  • It would improve the Question to specify which definition of integrability is being used, e.g. Riemann or Darboux, etc. – hardmath Jun 07 '16 at 23:54

1 Answers1

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Let $\mathcal{P}$ denote the family of all partitions of $[a,b].$

Then

$$\{L(f,Q_n): n \in \mathbb{N}\} \subset \{L(f,P): P \in \mathcal{P}\},\\ \{U(f,P_n): n \in \mathbb{N}\} \subset \{U(f,P): P \in \mathcal{P}\},$$

and

$$\sup_n L(f,Q_n) \leqslant \sup_P L(f,P) =\underline{\int}_a^b f\leqslant \overline{\int}_a^b f =\inf_P \,U(f,P) \leqslant \inf_n \,U(f,P_n)$$

Hence, $\sup_n L(f,Q_n) = \inf_n \,U(f,P_n) = I$ implies the lower and upper Darboux integrals are equal to $I$.

Therefore,

$$\int_a^bf = I$$

RRL
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