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of the functional F$_{y}$$_{|}$$_{x}$ on D ( W ) by 

<img src='content_image/316407.jpg'>

Thus we can define the conditional distribution F$_{y}$$_{|}$$_{x}$ as a generalized function in the space D $^{∗}$( W ) . 

When d$_{x}$ = 1 this is an exhaustive representation. When d$_{x}$ > 1 it may be advantageous to consider a derivative with respect to a d$_{x}$ − dimensional argument. Consider the conditioning vector, x , component-wise, and con- sider the multivariate copula function, C$_{F}$$_{x}$$_{1}$$_{,...,F}$$_{x}$$_{d}$$_{,F}$$_{y}$ ( F$_{x}$$_{1}$, ..., F$_{x}$$_{d}$, F$_{y}$ ) ; to sim- plify notation we drop the subscript to denote it simply by C. Then by a similar argument for any ψ ∈ D ( W ) where W = (0 , 1) d$_{x}$ we obtain ( F$_{y}$$_{|}$$_{x}$, ψ ) = ( − 1) d$_{x}$ ( C, ∂ $^{d$_{x}$}$ψ ) = 

<img src='content_image/316408.jpg'>

Remark 1. Similarly to Corollary 1, the generalized function F$_{y}$$_{|}$$_{x}$ can be extended as a linear continuous functional from being defined on the space D ( W ) of infinitely differentiable functions to a linear continuous functional defined by (11) on any space D$_{k}$ ( W ) with k ≥ 1 and for ( ?? ) to D$_{k}$ ( W ) for the corresponding W and k ≥ d$_{x}$. 

Remark 2. If the function C were suitably differentiable the functional ( F$_{y}$$_{|}$$_{x}$, ψ ) would be defined for any continuous ψ with bounded support, that 

