> [!NOTE] Definition (Upper Riemann Sum) > Let $[a,b]$ be a [[Closed Real Interval|closed real interval]]. > > Let $f:[a,b]\to \mathbb{R}$ be a [[Bounded Real Function|bounded]] [[Real Function|real function]]. > >The *lower Riemann Integral* of $f$ over $[a,b]$ is given by $\underline{\int} f = \sup_{P}L(f,P)$where $\sup_{P}L(f,P)$ is the [[Supremum of Set of Real Numbers|supremum]] of the set of [[Lower Darboux Sum|lower Riemann sums]] of $f$ with respect to all [[Finite Partition of Closed Real Interval|finite partitions]] of $[a,b].$ >