Extending Coarse-Grained Measures
In  it is proved that a measure on a finite coarse-grained space extends, as a signed measure, over the entire power algebra. In  this result is reproved and further improved. Both the articles  and  use the proof techniques of linear spaces (i.e. they use multiplication by real scalars). In this note we show that all the results cited above can be relatively easily obtained by the Horn–Tarski extension technique in a purely combinatorial manner. We also characterize the pure measures and settle the dimension of the normalized-measure space. We then comment on a consequence of the results for circulant matrices. Finally, we take up the case of circle coarse-grained space and also establish a measure-extension result.