Theories {measure theory} can find discontinuous-function magnitudes {mathematical measure} {measure, mathematics}, for quantum mechanics, statistics, and probability.
process
Enclose set points in an open-set interval inside a finite or countably infinite set of non-overlapping intervals {union of non-overlapping denumerable open intervals}. To obtain lower bound {exterior measure}, sum non-overlapping intervals. Use sum to find set-point complement {interior measure}.
measure
If function has bound and is measurable, length, area, or volume is greatest lower bound {greatest exterior measure} and equals least upper bound {least interior measure}.
types
Boolean sigma-algebra can represent discontinuous-function measures. In intervals, Lebesgue generalized ordinary integrals, over discontinuous-function points, can find function values {P-measure}. If Lebesgue integrals are constant, P-measures {Lebesgue measure} are constant.
Point-set {measurable set} exterior measure can equal interior measure. If functions are greater than a number, and point sets are measurable, functions are measurable.
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Date Modified: 2022.0225