  We study the quantum measurement problem in the context of an infinite,
statistically uniform space, as could be generated by eternal inflation. 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 infinite space actually renders the Born rule
redundant, by physically realizing all outcomes of a quantum measurement in
different 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 oft-discussed
quantum frequency operator. This analysis unifies 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. It also
shows how, even for a single measurement, probabilities may be interpreted as
relative frequencies in unitary (Everettian) quantum mechanics. We also argue
that after discarding a zero-norm 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 unified 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 self-locate
in this collection.
