content_image/303237.jpg content_image/303236.jpg content_image/303235.jpg
is on the space D$_{0}$ ( W ) by ( ∂ $^{d$_{x}$}$C ( ..., . ) , ψ ) : 

<img src='content_image/303237.jpg'>

In the y argument the conditional distribution is an ordinary function so here y is considered just as a parameter of the generalized function. However, the definition of F$_{y}$$_{|}$$_{x}$ in (11) can be extended to a functional for functions defined on the product space; for any ψ$_{x,y}$ = ψ$_{x}$ ( x$_{1}$, ...x$_{d}$$_{x}$ ) ψ$_{y}$ ( y$_{1}$, ..., y$_{d}$$_{y}$ ) ∈ D ((0 , 1) $^{d$_{x}$}$) × D ( R $^{d$_{y}$}$) define the value of the functional by ( F$_{y}$$_{|}$$_{x}$, ψ$_{x,y}$ ) = 

<img src='content_image/303236.jpg'>

To define conditional density f$_{y}$$_{|}$$_{x}$ as a generalized function one would have 

<img src='content_image/303235.jpg'>

In general, the conditional distribution and conditional density depend on the conditioning variables, x, via the marginals, F$_{x}$ ; considering generalized functions makes this explicit. 

There are cases when the conditional distribution and conditional density are defined on the Euclidean space R $^{d$_{x}$}$. This is possible if the distribution function F$_{x}$ is strictly monotone in each argument; then the corresponding generalized density function is positive, moreover, since a monotone function 

