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## 1 Introduction 

A probability distribution function, F, that corresponds to a Borel measure on a Euclidean space R k (or its subspace) is always defined in the space of bounded functions. It can be viewed as the right-hand side of an integral equation: 

where the density represents the solution to the inverse problem 

<img src='content_image/298483.jpg'>

<img src='content_image/298484.jpg'>

Here I represents an integration operator for R $^{k}$: I ( f ) ( x ) = ∫ x 1 $_{−∞}$... ∫ x$_{k}$ $_{−∞}$f ( w ) dw$_{1}$...dw$_{k}$ and ∂ k = ∂ k ∂x 1 ...∂x$_{k}$ the inverse differentiation operator. 

When does the solution to the inverse problem exist? 

In the usual approach the integral operator I is assumed to operate on the space of integrable functions, e.g. L$_{1}$ (absolutely integrable functions) or L$_{2}$ (square integrable functions), - see e.g. Devroye and Gyorfi (1985), Carrasco, Florens, Renault (2007). The operator I maps density functions in L$_{1}$ into the space of absolutely continuous distribution functions. In this case the inverse operator ∂ k is defined and the inverse problem has a unique solution. 

The property of well-posedness requires that the solution continuously depend on the right-hand side function, in other words, if distribution func- 

