Uniqueness of entropy solution

Question

I am starting to study hyperbolic equations of conservation laws. I have read that there exists a unique entropy solution of the problem $$ \left\{\begin{array}{l} u_t+f(u)_x=0~,~~(x,t)\in\mathbb{R}\times(0,\infty)\\ u(x,0)=u_0(x)~,~~x\in\mathbb{R} \end{array}\right. $$ and it satisfies, for example: $$ \left|\left|u(\cdot,t)\right|\right|_{L_1}\le\left|\left|u_0\right|\right|_{L_1}~~\forall t>0. $$ I think that a function equal a.e. to an entropy solution is also an entropy solution, is it right? Then, how is it possible to obtain the previous bound if the line $\mathbb{R}\times\{0\}$ is a null space in the domain?