  In this paper, we aim to establish foundations of measurement theory in local
quantum physics. For this purpose, we discuss a representation theory of
completely positive (CP) instruments on arbitrary von Neumann algebras. We
introduce a condition called the normal extension property (NEP) and establish
a one-to-one correspondence between CP instruments with the NEP and statistical
equivalence classes of measuring processes. We show that every CP instrument on
an atomic von Neumann algebra has the NEP, extending the well-known result for
type I factors. Moreover, we show that every CP instrument on an injective von
Neumann algebra is approximated by CP instruments with the NEP. The concept of
posterior states is also discussed to show that the NEP is equivalent to the
existence of a strongly measurable family of posterior states for every normal
state. Two examples of CP instruments without the NEP are obtained from this
result. It is thus concluded that in local quantum physics not every CP
instrument represents a measuring process, but in most of physically relevant
cases every CP instrument can be realized by a measuring process within
arbitrary error limits, as every approximately finite dimensional (AFD) von
Neumann algebra on a separable Hilbert space is injective. To conclude the
paper, the concept of local measurement in algebraic quantum field theory is
examined in our framework. In the setting of the Doplicher-Haag-Roberts and
Doplicher-Roberts (DHR-DR) theory describing local excitations, we show that an
instrument on a local algebra can be extended to a local instrument on the
global algebra if and only if it is a CP instrument with the NEP, provided that
the split property holds for the net of local algebras.
