  We show that the Christoffel function (CF) factorizes (or can be
disintegrated) as the product of two Christoffel functions, one associated with
the marginal and the another related to the conditional distribution, in the
spirit of "the CF of the disintegration is the disintegration of the CFs". In
the proof one uses an apparently overlooked property (but interesting in its
own) which states that any sum-of-squares polynomial is the Christoffel
function of some linear form (with a representing measure in the univariate
case). The same is true for the convex cone of polynomials that are positive on
a basic semi-algebraic set. This interpretation of the CF establishes another
bridge between polynomials optimization and orthogonal polynomials.
