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 $\mu(B)=\infty$?