Image and Reciprocal Image of a Measure. Compatibility Theorem
Abstract
Description
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 $ϕ$ is defined between two measurable spaces, to each measure $μ$ introduced on the first space, there corresponds an image $ϕ[μ]$ on the second space, and, reciprocally, to each measure $ν$ defined on the second space the corresponds a reciprocal image $ϕ^{-1}[ν]$ on the first space. As the intersection $\cap$ of two measures is easy to introduce, a relation like $ ϕ[ μ\cap ϕ^{-1} [ν] ] = ϕ[μ] \cap ν$ makes sense. It is, indeed, a theorem of the theory. This theorem gives mathematical consistency to inferences drawn from physical measurements.
19 pages, 1 figure
19 pages, 1 figure