I am learning the book "Optimal transport old and new" by Villani.
We have the following notion of Regularizing kernels.
My first question is: if this regularizing kernel is similar to mollifier. Specifically, I think if we let $\mathcal{Y}=\{0\}$, $\phi=K_1(x,0)$ and $\phi_\epsilon=\epsilon^{-d}\phi(x/\epsilon)$, then the $\phi$ is indeed a mollifier.
My second question is: if we construct a mollifier like I described above, then is this $K_\epsilon\mu$ the mollification of $\mu$?
My third question is about this property: if $\mu$ is a finite measure supported in $\mathcal{Y}$, then $(K_\epsilon\mu)\nu$ converges to $\mu$ in duality with $\mathcal{C}_b(X)$. But if we let $\mathcal{Y}=\{0\}$, this $\mu$ can only be a Dirac measure on $0$. Is this convergence property true for measure $\mu$ without compact support?
Thank you!