# Monotonic set function

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Sometimes called **Monotone set function**.

## Definition

A set function [ilmath]f:E\rightarrow [0,\infty]\subset\mathbb{R} [/ilmath] is **monotonic**^{[1]} if for [ilmath]A,B\in E[/ilmath] we have [ilmath]A\subseteq B\implies f(A)\le f(B) [/ilmath]

We do not require a function that maps to [ilmath][0,\infty][/ilmath] any linearly ordered set will do. It will likely be encountered in this form though.

## See also

## References

- ↑ p37 - Halmos, Measure Theory, Springer Texts in mathematics - book 18