  Quantum mechanics introduces the concept of probability at the fundamental
level, yielding the measurement problem. On the other hand, recent progress in
cosmology has led to the "multiverse" picture, in which our observed universe
is only one of the many, bringing an apparent arbitrariness in defining
probabilities, called the measure problem.
  In this paper, we discuss how these two problems are intimately related with
each other, developing a complete picture for quantum measurement and
cosmological histories in the quantum mechanical universe. On one hand, quantum
mechanics eliminates the arbitrariness of defining probabilities in the
multiverse, as discussed in arXiv:1104.2324. On the other hand, the multiverse
allows for understanding why we observe an ordered world obeying consistent
laws of physics, by providing an infinite-dimensional Hilbert space. This
results in the irreversibility of quantum measurement, despite the fact that
the evolution of the multiverse state is unitary.
  In order to describe the cosmological dynamics correctly, we need to identify
the structure of the Hilbert space for a system with gravity. We argue that in
order to keep spacetime locality, the Hilbert space for dynamical spacetime
must be defined only in restricted spacetime regions: in and on the (stretched)
apparent horizon as viewed from a fixed reference frame. This requirement
arises from eliminating all the redundancies and overcountings in a general
relativistic, global spacetime description of nature. It is responsible for
horizon complementarity as well as the "observer dependence" of
horizons/spacetime---these phenomena arise to represent changes of the
reference frame in the relevant Hilbert space. This can be viewed as an
extension of the Poincare transformation in the quantum gravitational context,
as the Lorentz transformation is viewed as an extension of the Galilean
transformation.
