  In technical terms, where \(U\) is a utility function normalized so that \(U(\Omega) = 0\), \(inf\) is the greatest lower bound of the values assigned by \(U\), \(sup\) is the least upper bound of the values assigned by by \(U\), and \(\lambda\) is a parameter falling between \(-1/inf\) and \(-1/sup\), the fractional linear transformation \(\langle P_{\lambda}, U_{\lambda} \rangle\) of \(\langle P, U \rangle\) corresponding to \(\lambda\) is given by: