We consider the following probability measure on $\mathbb{R}^2$: $\mu = Leb\vert_{[0,1]} \times \delta_0$. Furthermore the following dilation, say $d$, is defined as $(x,0) \mapsto \frac{1}{2}(\delta_{(x,f(x))}+\delta_{(x,-f(x))})$ where $f$ is a non-continuous function function. Define the probability measure $\nu$ as $\nu = \mu d$. Why is this kernel/dilation not a Lipschitz kernel?
Short remark about Lipschitz kernel: A kernel $d:x\mapsto \theta_x$ (disintegration) transporting $\mu$ to $\mu d$ is called Lipschitz if there exists a subset of $\mathbb{R}^d$ of full $\mu$-measure such that $d$ restricted to this subset is Lipschitz of constant $1$ from $(\mathbb{R}^d,\vert\vert \cdot\vert\vert_{\mathbb{R}^d}$) to $(\mathcal{P}(\mathbb{R}^d),W)$, where $\mathcal{P}(\mathbb{R}^d)$ are the probability measures on $\mathbb{R}^d$ and $W$ is the Kantorovich distance between $\theta$ and $\theta'$ defined by
$W(\theta, \theta') = \sup_{f\text{ } 1-\text{Lipschitz}} \vert\vert \int f d\theta - \int f d\theta'\vert\vert_{\mathbb{R}^d}$.
