**Abstract:** Indirect measurements were present in the statistical structure of
QM from its first development by von Neumann, and their basic mathematically
theory was later described by Kraus. Under the name quantum non-demolition
measurements, the theory became influential both experimentally and as a toy
model for measurement process in quantum mechanics. A recent example of the
former is the photon counting experiment of Haroche. The main aim of
the lecture would be to introduce the theory and to describe its richness in
connections to several mathematical structures: C* algebras, the estimation
theory, equivalence of ensembles, martingales, etc. The talk is based on a
joint work with M. Ballesteros, J. Frohlich and B. Schubnel.