Abstract
This four-part paper stems from previous work of certain of the authors, where the issue of inducing distributions on lower dimensional spaces arose as a natural outgrowth of the main goal: the estimation of conditional probabilities, given other partially specified conditional probabilities as a premise set in a probability logic framework. This paper is concerned with the following problem. Let 1 m < n be fixed positive integers, 
some open domain, and 
a function yielding a full partitioning of D into a family, denoted M(h), of lower-dimensional surfaces/manifolds via inverse mapping h
1 as D
M(h), where M(h)
d
{h
1(t) : t in range(h)}, noting each h
1(t) can also be considered the solution set of all X in D of the simultaneous equations h(X) t. Let X be a random vector (rv) over D having a probability density function (pdf) . Then, if we add sufficient smoothness conditions concerning the behavior of h (continuous differentiability, full rank Jacobian matrix dh(X)/dX over D, etc.), can an explicit elementary approach be found for inducing from the full absolutely continuous distribution of X over D a necessarily singular distribution for X restricted to be over M(h) that satisfies a list of natural desirable properties? More generally, for fixed positive integer r, we can pose a similar question concerning rv (X), when 
is some bounded a.e. continuous function, not necessarily admitting a pdf.
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