L-\(\sigma\)-algebras and L-measures (Q1181994)

A new binary operation \(\underline .\) on the class \(F(X,L)\) of \(L\)- subsets of \(X\) has been introduced in the following way: \[ (f\underline . g)(x)=\begin{cases} f(x), & \text{ if } f(x)\neq g(x),\\ 0, & \text{ if } f(x)=g(x), \end{cases} \] for every \(f,g\in F(X,L)\). The main result in the paper proves that, under a suitable hypothesis, an \(L\)-\(\sigma\)-algebra \(\tau\) is generated if and only if it is closed with respect to \(\underline .\), where by generated \(L\)-\(\sigma\)-algebra the authors mean the following: an \(L\)-\(\sigma\)-algebra \(\tau\) is called generated if there exists a classical \(\sigma\)-algebra \({\mathcal U}\) on \(X\) such that \(f\in\tau\) if and only if \(f\) is measurable with respect to \({\mathcal U}\).