upper integral

upper integral

[′əp·ər ′int·ə·grəl]
(mathematics)
The upper Riemann integral for a real-valued function ƒ(x) on an interval is computed to be the infimum of all finite sums over all partitions of the interval, the sums having terms given by (xi -xi-1) yi , where the xi are from a partition, and yi is the largest value of ƒ(x) over the interval from xi-1to xi .
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The contributions that make up the main body of the text are devoted to the value of the first order of singular optimization problems, weak convergence in metric spaces, mappings with upper integral bounds forp-moduli, and many other subjects related to nonlinear analysis and optimization.
The supremum--which is known to be smaller than or equal to the desired integral [integral] f (x) dx--is called the lower integral, and ther infimum--which is known to be larger than or equal to the desired integral [integral] f (x) dx--is called the upper integral.
For the expected value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the corresponding lower and upper integrals are called lower and upper expected values, and denoted by [E.