Prove that if $\forall \epsilon > 0 \; \exists$ integrable functions $h,g$ at $[a,b]$ s.t $h\leq f \leq g $ and $\int \limits_{a}^{b}(g-f) < \epsilon$ $\implies f$ is integrable.

We've discussed both Riemann and Darboux integrals, and defined integral using Darboux sums. I've tried to use the definition that $f$ is integrable $\iff$ $\forall \epsilon > 0 \; \exists$ step funcions $\phi$ and $\psi$, $\phi \leq f \leq \psi$ s.t $\int \limits_a^b f-\phi < \epsilon$ and $\int \limits_a^b \psi-f < \epsilon$. However I didn't manage to define either $\phi$ or $\psi$ that will do the job. Any ideas?

  • $\begingroup$ There is something wrong in your formulation. $g$ appears too many times. $\endgroup$ – uniquesolution Apr 15 '17 at 13:13
  • $\begingroup$ @uniquesolution thanks, corrected. $\endgroup$ – blz Apr 15 '17 at 13:14
  • $\begingroup$ @uniquesolution yes $\endgroup$ – blz Apr 15 '17 at 13:16
  • $\begingroup$ @uniquesolution you're right, of course. I think they were expecting us to prove it using more basic definitions of integrability, such as Darboux sums, or using step functions. $\endgroup$ – blz Apr 15 '17 at 13:20

Let $P=\{x_0=a,x_1.......x_n=b\}$ a partition of $[a,b]$

Take $\phi(x)=\sum_{k=1}^nl_k\mathbb{X}_{[x_{k-1},x_k]}(x)$ where $l_k=\inf\{f(x):x \in [x_{k-1},x_k]\}$

And $\psi(x)=\sum_{k=1}^nu_k\mathbb{X}_{[x_{k-1},x_k]}(x)$ where $u_k=\sup\{f(x):x \in [x_{k-1},x_k]\}$

These two functions are simple functions..

$\mathbb{X}_A$ is the indicator function of the set $A$

You can continue from here..

  • $\begingroup$ Why, for example, does $int \limits_a^b \psi-f < \epsilon$? $\endgroup$ – blz Apr 15 '17 at 15:23
  • $\begingroup$ because of the integrability of $f$ ,if yoy assume that $f$ is integrable $\endgroup$ – Marios Gretsas Apr 15 '17 at 15:24
  • $\begingroup$ the integral of these two functions ,i defined are the lower and upper Darboux sums respectively $\endgroup$ – Marios Gretsas Apr 15 '17 at 15:25
  • $\begingroup$ The value of each of the indicator functions is the length of the interval $\endgroup$ – Marios Gretsas Apr 15 '17 at 15:26

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