6
$\begingroup$

Let $(E,d)$ be a metric space, $\mathscr E$ be its Borel $\sigma$-algebra and $\mu$ be a $\sigma$-finite measure on $(E,\mathscr E)$. Let the function $p:E\times E\to\mathbb R_+$ be non-negative and jointly measurabe: $p\in\mathscr E\otimes \mathscr E$. Let's assume that for any compact set $A\subset E$ there is a constant $\lambda_A$ such that $$ |p(x'',y) - p(x',y)|\leq \lambda_A\cdot d(x',x'')\text{ for all }x',x''\in A,y\in E \tag{1} $$ and let us assume that $$ \int\limits_E p(x,y)\mu(\mathrm dy) = 1 \tag{2}\text{ for all }x\in E. $$

Clearly, if $A$ is compact, then $P(x,B):=\int\limits_B p(x,y)\mu(\mathrm dy)$ is Lipschitz on $A$ whenever $\mu(B)<\infty$: $$ |P(x'',B) - P(x',B)|\leq \lambda_A\cdot\mu(B)d(x'',x'). $$ Does the Lipschitz continuity of $P(x,B)$ also hold on $A$ if $\mu(B)=\infty$?

  • 0
    The last displayed line should be $|P(x'',B)-\cdots|$? It's true if the measure of $B$ or $B^c$ is finite.2012-05-21
  • 0
    @DavideGiraudo: thanks, that was a typo. Yeah, that's true if $B^c$ is of finite measure because $P(x,E) \equiv 1$. But I wonder if there is a counterexample for some $\mu(B) = \infty$2012-05-21
  • 0
    @Norbert I guess the OP meant the Lipschitz condition on compact subsets.2012-05-21
  • 0
    @DavideGiraudo, if $\mu(B)=+\infty$, the last inequality is allways holds, isn't it?2012-05-21
  • 0
    I agree, but the problem is to determine whether the map $x\mapsto P(x,B)$ is Lipschitz continuous over compact sets when $0<\mu(B)<\infty$. I agree that it's ambiguous.2012-05-21
  • 0
    @Davide could you tell me, if it is more clear now?2012-05-21
  • 0
    You mean Lipschitz continuity over compact sets I guess.2012-05-21
  • 0
    @Davide yes. Do you have ny idea how to prove/disprove it?2012-05-21
  • 0
    I've not the answer, it works when the topology is discrete since $P(\cdot,B)$ is bounded. But I don't know in the general case.2012-05-21
  • 0
    @Davide isn't any function locally Lipschitz in discrete topology, since the only compact sets are finite sets?2012-05-22

1 Answers 1