content_image/345634.jpg content_image/345633.jpg content_image/345632.jpg content_image/345631.jpg content_image/345630.jpg
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$^{α}$〉 $_{0}$. 

<img src='content_image/345634.jpg'>

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 

<img src='content_image/345633.jpg'>

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), 

<img src='content_image/345632.jpg'>

where 

for unit eigenstates | s $^{α}$, e$^{αβ}$, c$^{αβ}$〉 $_{0}$. 

<img src='content_image/345631.jpg'>

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 

<img src='content_image/345630.jpg'>

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].) 

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. 

