where the | s $^{α}$〉 0 are defined as in Eq. (30) and the | s $^{α}$, e$^{αβ}$〉 0 are unit eigenstates. Since, in the present formalism, all states are ultimately defined on the infinite-dimensional space of fine-grained continuous position states, such a discrete subdivision is always possible. Then which respects the unit normalization of the | s $^{α}$, e$^{α}$〉 $_{0}$. Introduce an ancillary system in a hypervolume C separate from V , but able to interact with E without influencing the interaction of E with S . The interaction between C and E is such that for an appropriate set of projection operators ˆ P αβ C indexed parallel to the ˆ P αβ E . Then, using Eqs. (37), (38) and (39) with Eq. (34), where for unit eigenstates | s $^{α}$, e$^{αβ}$, c$^{αβ}$〉 $_{0}$. The terms in Eq. (40) all now have equal coefficients, so we take the corresponding states to all be equally likely. Since there a total of M terms, the probability of any one of the states is 1 /M . Further, since, for each α , m$_{α}$ of the overall system/environment/ancilla states correspond to the system outcome α , the probability for this outcome is Of course, this argument only establishes the Born rule for the overall states | s $^{α}$, e$^{α}$〉 $_{0}$. But [24] establishes that, if Born’s rule holds for such states, then it follows that the statistics of repeated measurement experiments would be expected to also follow this rule. which is just Born’s law. By continuity, the same conclusion can be extended to all real ψ α $_{S}$. (Note that the derivation here also assumes the additivity of probabilities, but it is possible to come to the same conclusion without making this assumption [32].) 25 p$_{α}$ = m$_{α}$/M = | ψ α S | $^{2}$, ˆ G$_{C}$ | s $^{α}$, e$^{αβ}$〉 0 = | s $^{α}$, e$^{αβ}$, c$^{αβ}$〉 $_{0}$, ˆ G$_{C}$ ˆ G$_{V}$ | Ψ$_{I}$ 〉 0 = ∑ α √ m$_{α}$/M m$_{α}$ ∑ β =1 ˆ G$_{C}$ | s $^{α}$, e$^{αβ}$〉 $_{0}$/ √ m$_{α}$ = αβ 1 /M | s $^{α}$, e$^{αβ}$, c$^{αβ}$〉 $_{0}$, (40) ˆ G$_{C}$ ˆ P αβ E = δ$_{αγ}$δ$_{βζ}$ ˆ P γζ C ˆ G$_{C}$ ˆ P αβ E , (39) | s $^{α}$, e$^{α}$〉 0 = ˆ P α E | s $^{α}$〉 0 = m$_{α}$ ∑ β =1 | s $^{α}$, e$^{αβ}$〉 $_{0}$/ √ m$_{α}$ , (38)