Fuzzy measure

Remarks

Often $\mathcal{C}$ is a $\sigma$-algebra or the set of all subsets of a finite set $X$. Example: let $\mu$ be a probability measure on a $\sigma$-algebra $\mathcal{C}$ and $g(E) = \mu(E)^2$. Then $g(\emptyset) = 0$, and $g$ is monotone because $\mu$ is monotone and squaring is increasing on $[0, 1]$. It is not additive: if $\mu(E) = \mu(F) = 1/2$ with $E \cap F = \emptyset$, then $g(E) + g(F) = 1/2$ but $g(E \cup F) = 1$. Non-example: $g(E) = \mu(E) - \mu(E)^2$ vanishes on $\emptyset$, but it is not monotone: $g(E) = 1/4$ for $\mu(E) = 1/2$, while $g(X) = 0$. Every measure is a fuzzy measure.