Image and Reciprocal Image of a Measure. Compatibility Theorem

Abstract

It is proposed that to the usual probability theory, three definitions and a new theorem are added, the resulting theory allows one to displace the central role usually given to the notion of conditional probability. When a mapping $\phi$ is defined between two measurable spaces, to each measure $\mu$ introduced on the first space, there corresponds an image $\phi[\mu]$ on the second space, and, reciprocally, to each measure $\nu$ defined on the second space the corresponds a reciprocal image $\phi^{-1}[\nu]$ on the first space. As the intersection $\cap$ of two measures is easy to introduce, a relation like $ \phi[ \mu \cap \phi^{-1} [\nu] ] = \phi[\mu] \cap \nu $ makes sense. It is, indeed, a theorem of the theory. This theorem gives mathematical consistency to inferences drawn from physical measurements.