Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
well, yes: the zero measure. –  Mark Aug 11 '11 at 11:03
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.