Family of measure that admits a continuous density

This question is a generalization of an example provided in Absolute continuous family of measures.

Consider a metric space $(X,\rho)$ with a Borel $\sigma$-algebra $\mathcal B(X)$. Consider a family of probability measures $K:X\times \mathcal B(X)\to[0,1]$ such that

1. $x\mapsto K(x,B)$ is a measurable function for each $B\in\mathcal B(X)$ and
2. $B\mapsto K(x,B)$ is a probability measure on $(X,\mathcal B(X)).$

When does exist such a measure $\mu$ such that $\mu(\cdot)\gg K(x,\cdot)$ for each $x\in X$ and $$\xi(x,y):=\frac{\mathrm dK(x,\cdot)}{\mathrm d \mu(\cdot)}$$ is a continuous/Lipschitz continuous function?

First thoughts are the following: necessary conditions are continuity of $K(x,B)$ for each bounded $B$ (for the continuity of $\xi$) and Lipschitz continuity of $K(x,B')$ for each $B'$ s.t. $\mu(B')<\infty$ (for the Lipschitz continuity of $\xi$). However, the last one is using the measure $\mu$ which we can also choose.

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well, yes: the zero measure. – Mark Aug 11 '11 at 11:03