The random set and the cutting of random fuzzy sets (Q1278852)

The author presents several (rather obvious) results that relate a (possibly random) function \(A:X\to[0,1]\) (also called a fuzzy set), its random threshold \(A_\alpha= \{x\in X:A(x)\geq\alpha\}\) at random level \(\alpha\) (also called an \(\alpha\)-cut) and the coverage probabilities \(\mathbb{P}\{x\in A_\alpha\}\), \(x\in X\).