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I know if there exists a measurable partition $P$ of $[a,b]$, $f$ is Lebesgue integrable on $[a,b]$ when $$ \inf_P U[f;P] = \sup_P L[f;P]$$ where the infemum and supremum are over all measurable partitions.

However, what can I say the function f if $U[f;P]$=$L[f;P]$?

Caleb Stanford
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user140744
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  • Are $U[f;P]$ and $L[f;P]$ the infemum and supremum over all measurable partitions $P$? If so, this is poor notation. – Caleb Stanford May 06 '14 at 04:47
  • Aha! @glacier has messed up the notation. I will fix it. – Caleb Stanford May 06 '14 at 07:48
  • @OP Any idea of your own? Frankly, I am surprised that none comes to anybody who knows what upper/lower Riemann sums are... You might want to draw a figure. – Did May 06 '14 at 08:57
  • @OP It will help to prove a small lemma first: Suppose $E$ is equipped with the Lebesgue measure, and let $f$ be a bounded function on some compact set $O\subset E$. Show that if $\inf_{x\in O} f(x)=\sup_{x\in O} f(x)$ then $f$ is constant on $O$. – Sawyer Mar 20 '17 at 06:33

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