Abstract: {For $1 < p < \infty$, let $S(L_p)_+$ be the set of positive elements in $L_p$ with norm one. Assume that $V_0: S(L_p(\Omega_1))_{+}\to S(L_p(\Omega_2))_{+}$ is a surjective norm-additive map; that is, \[\|V_0(x)+V_0(y)\|=\|x+y\|,\quad\forall\,x, y\in S(L_p(\Omega_1 ))_{+}.\] In this paper, we show that $V_0$ can be extended to an isometry from $L_p(\Omega_1)$ onto $L_p(\Omega_2)$.

Key words: Norm-additive mappings, positive cones, L_p spaces