  This paper proposes an extension of Chaitin's halting probability \Omega to a
measurement operator in an infinite dimensional quantum system. Chaitin's
\Omega is defined as the probability that the universal self-delimiting Turing
machine U halts, and plays a central role in the development of algorithmic
information theory. In the theory, there are two equivalent ways to define the
program-size complexity H(s) of a given finite binary string s. In the standard
way, H(s) is defined as the length of the shortest input string for U to output
s. In the other way, the so-called universal probability m is introduced first,
and then H(s) is defined as -log_2 m(s) without reference to the concept of
program-size.
  Mathematically, the statistics of outcomes in a quantum measurement are
described by a positive operator-valued measure (POVM) in the most general
setting. Based on the theory of computability structures on a Banach space
developed by Pour-El and Richards, we extend the universal probability to an
analogue of POVM in an infinite dimensional quantum system, called a universal
semi-POVM. We also give another characterization of Chaitin's \Omega numbers by
universal probabilities. Then, based on this characterization, we propose to
define an extension of \Omega as a sum of the POVM elements of a universal
semi-POVM. The validity of this definition is discussed.
  In what follows, we introduce an operator version \hat{H}(s) of H(s) in a
Hilbert space of infinite dimension using a universal semi-POVM, and study its
properties.
