Definition:Lower Sum

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Definition

Let $\closedint a b$ be a closed real interval.

Let $f: \closedint a b \to \R$ be a bounded real function.


Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$.

For all $\nu \in \set {1, 2, \ldots, n}$, let $m_\nu^{\paren f}$ be the infimum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$.


Then:

$\displaystyle \map {L^{\paren f} } P = \sum_{\nu \mathop = 1}^n m_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$

is called the lower sum of $f$ on $\closedint a b$ belonging (or with respect) to (the subdivision) $P$.


If there is no ambiguity as to what function is under discussion, $m_\nu$ and $\map L P$ are usually used.


Also known as

The notation $\map L {P, f}$ can be used in place of $\map {L^{\paren f} } P$.


The lower sum is also known as the lower Darboux sum or the lower Riemann sum.


Also see


Sources