The density ratio is undefined on zero marginal fibres

The density p is the indicator of a Borel rectangle. By [F3]–[F5], ${\int }_0^{1}1dx=1$; endpoints have measure zero by containment in intervals with arbitrarily small lengths. Tonelli [F6] gives total joint mass $1\cdot 1=1$ and marginal $m(y)={\mathbf{1}}_{(0,1)}(y)$. The measure construction is [F7]. At y=2, $p(x,2)=0$ for every x and $m(2)=0$. The asserted quotient is therefore 0/0, which is undefined, at every x on that fibre. Thus the claimed everywhere formula fails for a fully normalized bounded joint density.

Put $\rho (A)={\lambda }_1(A\cap (0,1))$ and $K(y,A)=\rho (A)$ for all real y. By [F7] and the mass calculation, rho is a probability, and constant evaluations are measurable, proving [F2]. On $0<y<1$ this agrees with the density ratio. On its complement it is the supplied filler allowed by [F1]. Directly, for Borel A,B, ${\int }_{B}K(y,A)P_Y(dy)={\lambda }_1(A\cap (0,1)){\lambda }_1(B\cap (0,1))=P(X\in A,Y\in B),$ with the last equality from [F6]. So the extension is a conditional law, including an everywhere probability section at y=2. For example $K(2,(0,1\mathrm{/}2))=1\mathrm{/}2$, a chosen valid extension value, not a value of 0/0.