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on ( x, y ) that satisfies Assumption 2 the class of functions g is restricted. 

Assumption 3. The function g is continuously differentiable with respect to the differentiation operator ∂ $^{d$_{y}$}$. 

Any polynomial function satisfies Assumption 3, and thus conditional mean of y, or conditional variance (if they exist) can be considered. If the function were not to satisfy the differentiability assumption, the class of distributions would need to be correspondingly restricted. 

Consider D ( R d$_{y}$ ) and a locally finite partition of unity on R d$_{y}$ by a set of suitable functions, ”bump” functions from D ( R d$_{y}$ ) : { ψ$_{ν}$ } , where ψ$_{ν}$ ∈ D ( R d$_{y}$ ) , ψ ≥ 0 and Σ$_{ν}$ ψ$_{ν}$ ( y ) ≡ 1; also any y can belong to support of only a finite number of ψ$_{v}$. See e.g. Gel’fand and Shilov, 1964, v.1, p.142 for a construction. 

Then define ( gf$_{y}$$_{|}$$_{x}$, ψ$_{ν}$ ) = ∫ g ( y ) f$_{y}$$_{|}$$_{x}$ ( x, y ) ψ$_{v}$ ( y ) dy ; under Assumption 3 this expression is (as usual integrating by parts and using boundedness of support of ψ$_{v}$ ): 

<img src='content_image/321893.jpg'>

This expression represents a generalized function on D ( W ) given for any 

