  The Leggett-Garg inequality is a widely used test of the "quantumness" of a
system, and involves correlations between measurements realized at different
times. According to its widespread interpretation, a violation of the
Legget-Garg inequality disproofs macroscopic realism and non-invasiveness.
Nevertheless, recent results point out that macroscopic realism is a model
dependent notion and that one should always be able to attribute to
invasiveness a violation of a Legget-Garg inequality. This opens some natural
questions: how to provide such an attribution in a systematic way? How can
apparent macroscopic realism violation be recast into a dimensional independent
invasiveness model? The present work answers these questions by introducing an
operational model where the effects of invasiveness are controllable through a
parameter associated with what is called the {\it measurability} of the
physical system. Such a parameter leads to different generalized measurements
that can be associated with the dimensionality of a system, to measurement
errors or to back action.
