Initial Rest Condition for LCCDE causal LTI systems
If you are given an input $x(t)$ with $x(t)=0$ for $t&lt;t_0$, and you specify an initial condition $y(t_1)=0$ for $t_1&gt;t_0$, then the resulting system is generally non-causal, because we already know the system's response at $t_1&gt;t_0$, regardless of the input signal in the interval $[t_0,t_1]$.

For a system to be linear (in the sense used by Oppenheim) and causal, we need to specify the initial condition at the point $t_0^-$, i.e., we require that the left-sided limit $y(t_0^-)$ satisfies $y(t_0^-)=0$.

For the given example, if an impulse is applied at $t=0$, we require $y(0^-)=0$ for the system to be linear and causal. Note that there is no requirement on the right-sided limit $y(0^+)$.

Clearly, the response $y(t)=e^{-at}u(t)$ satisfies the initial condition $y(0^-)=0$.