ψ α S S (Ψ$_{I}$) = 0, then | s α S (Ψ$_{I}$) 〉 0 can be chosen arbitrarily from the eigenspace of ˆ P α S S .) Then, because of Eq. (27), Thus, where | s α S , a α S 〉 0 = ˆ G$_{A}$ | s α S 〉 0 (and the explicit dependence on Ψ$_{I}$ has been dropped for simplicity of notation). Because of Eq. (28), a similar relationship to Eq. (31) holds between E and A . Therefore, using this and Eq. (32) in Eq. (29) gives where | s α S , a α S , e α S 〉 0 = ˆ G$_{E}$ | s α S , a α S 〉 $_{0}$. Equation (33) is essentially the form assumed for a measurement state in [24]. Each | s α S , a α S , e α S 〉 0 is a state of the overall system V with outcome α$_{S}$ for S and correlated outcomes for A and E . And these state are orthogonal, so Eq. (33) certainly represents a consistent family of decoherent histories. Thus, one clearly wants to interpret | ψ α S S | 2 as the probability for | s α S , a α S , e α S 〉 0 according to the Born rule. The next section turns, then, to establishing the usual Born-rule probability interpreta- tion. Note, though, that the derivation of Eq. (33) is independent of this interpretation. Consider now a hypervolume V bounded by times T$_{I}$ and T$_{F}$ and divided into a system S and its environment E , such that for appropriate projection operators ˆ P α E and ˆ P α $_{S}$. Then, following a similar argument to Sec. V, given an initial state | Ψ$_{I}$ 〉 $_{0}$, 21 ˆ G$_{V}$ | Ψ$_{I}$ 〉 0 = ∑ α ψ α S | s $^{α}$, e$^{α}$〉 $_{0}$. (34) ˆ G$_{V}$ = ˆ G$_{E}$ ˆ G$_{S}$ = ∑ α ˆ P α E ˆ G$_{E}$ ˆ P α S ˆ G$_{S}$ , VI. BORN’S RULE ˆ G$_{A}$ ˆ P α S S = ( ∑ α A ˆ P α A A ) ˆ G A ˆ P α S S = ˆ P α S A ˆ G$_{A}$ ˆ P α S S . (31) ˆ P α S A ˆ G$_{A}$ ˆ P α S S ˆ G$_{S}$ | Ψ$_{I}$ 〉 0 = ˆ G$_{A}$ ˆ P α S S ˆ G$_{S}$ | Ψ$_{I}$ 〉 0 = ψ α S S | s α S , a α S 〉 $_{0}$, (32) ˆ G$_{V}$ | Ψ$_{I}$ 〉 0 = ∑ α S > 0 ψ α S S | s α S , a α S , e α S 〉 $_{0}$, (33)