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## variables and conditional density in the space of generalized functions 

Conditioning is an awkward operation as discussed e.g. in Chang and Pollard (1997). Here the question posed is limited to conditioning on a variable or vector in a joint distribution, that is given a joint distribution function F$_{x,y}$ ( ., . ) on R d$_{x}$ × R d$_{y}$ define a (generalized) function F$_{y}$$_{|}$$_{x}$ ( ., . ) that represents the conditional distribution of y given x. A problem associated with such conditioning is that the conditional distribution function may not exist for every point x. 

Denote by F$_{x}$, F$_{y}$ the marginal distribution functions of x, y, correspond- ingly. 

Consider limits of ratios to define conditioning: 

<img src='content_image/312639.jpg'>

As discussed is numerous papers there is a problem defining such a limit (e.g. Pfazagle, 1979); here it will be demonstrated that the limit exists in a particular space of generalized functions. Assume that the distribution function F$_{x}$ is continuous; continuity of this distribution of course does not preclude singularity. 

Assumption 2. The marginal distribution function F$_{x}$ ( x ) is continuous on R $^{d$_{x}$}$. 

