As stated in Probabalistic Robotics, the proof for correctness of a Bayesian Filter relies on the fact that
$$p(x_{t-1}|z_{1:t-1},\ u_{1:t}) = p(x_{t-1}|z_{1:t-1},\ u_{1:t-1})$$
In order to justify this, they say
$u_t$ can be safely omitted ... for randomly chosen controls
Why is that required? Isn't that true because the control input at time step t cannot possibly effect the state at time t-1?