Let $u,v \in L^2(0,T;H^1_0)$ with $u_t, v_t \in L^2(0,T;H^{-1})$.
We know that $$t \mapsto (u(t), v(t))_{L^2}$$ is absolutely continuous after a change on a set of measure zero.
Do we always identify the above function with the absolutely continuous representative? What goes wrong if we don't?