Summary: The Fredholm theory of integral equations is applied to the Perron-Frobenius equation which determines the invariant measure of nonlinear difference equations. The resultant Fredholm determinant \(D\) is identical to \(1/\zeta\). where \(\zeta\) is the Artin-Mazur-Ruelle \(\zeta\)-function. From the investigation of \(D\) we get the observable condition of the invariant measure of chaos, its stability against external noise, etc.