Proposition: Sequential operational law

For specified instruments ${\mathcal{I}}^{(1)},\dots ,{\mathcal{I}}^{(k)}$,

is a probability law. The law depends on the instruments, not merely on a list of self-adjoint observables or their ordered Weyl expectations.

Proof. Complete positivity gives nonnegative outcome probabilities. Summing the last outcome applies a trace-preserving map. Repeat backwards through the sequence to obtain total probability $\mathrm{Tr}\rho =1$. Different state-update maps with the same effects can give different later probabilities.