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We study the quantum measurement problem in the context of an innite, statistically uniform
space, as could be generated by eternal in
ation. It has recently been argued that when identical
copies of a quantum measurement system exist, the standard projection operators and Born rule
method for calculating probabilities must be supplemented by estimates of relative frequencies of
observers. We argue that an innite space actually renders the Born rule redundant, by physically
realizing all outcomes of a quantum measurement in dierent regions, with relative frequencies given
by the square of the wave function amplitudes. Our formal argument hinges on properties of what
we term the quantum confusion operator, which projects onto the Hilbert subspace where the Born
rule fails, and we comment on its relation to the oftdiscussed quantum frequency operator. This
analysis unies the classical and quantum levels of parallel universes that have been discussed in
the literature, and has implications for several issues in quantum measurement theory. Replacing
the standard hypothetical ensemble of measurements repeated ad innitum by a concrete decohered
spatial collection of experiments carried out in dierent distant regions of space provides a natural
context for a statistical interpretation of quantum mechanics. It also shows how, even for a sin
gle measurement, probabilities may be interpreted as relative frequencies in unitary (Everettian)
quantum mechanics. We also argue that after discarding a zeronorm part of the wavefunction,
the remainder consists of a superposition of indistinguishable terms, so that arguably \collapse" of
the wavefunction is irrelevant, and the \many worlds" of Everett's interpretation are unied into
one. Finally, the analysis suggests a \cosmological interpretation" of quantum theory in which the
wave function describes the actual spatial collection of identical quantum systems, and quantum
uncertainty is attributable to the observer's inability to selflocate in this collection.
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