monotone function

A monotonic function is a function which is either entirely nonincreasing or nondecreasing. A function is monotonic if its first derivative (which need not be continuous) does not change sign. The term monotonic may also be used to describe set functions which map subsets of the domain to non-decreasing values of the codomain. In particular, if f:X->Y is a set function from a collection of sets X to an ordered set Y, then f is said to be monotone if whenever A⊆B as elements of X, f(A)<=f(B). This particular definition comes up frequently in measure theory where many of the families of functions defined (including outer measure, premeasure, and measure) begin by considering monotonic set functions.

completely monotonic function | measure | measure theory | monotone | monotone decreasing | monotone increasing | nondecreasing function | nonincreasing function | outer measure | premeasure