For each $x\in R^d$, let $\nu(x,dz)$ be a L\'evy measure, i.e., $$ \int_{R^d}(|z|^2\wedge1)\nu(x,dz)<\infty. $$ Let $\mu$ be a probability measure on $R^d$ such that $$ \int_{R^d}\int_{R^d}(|z|^2\wedge1)\nu(x,dz)\mu(dx)<\infty. $$ Can we find a sequence of $\nu_n(x,dz)$ such that $$ x\to\int_{R^d}(|z|^2\wedge1)\nu_n(x,dz) $$ is continuous and $$ \int_{R^d}\left|\int_{R^d}(|z|^2\wedge1)\nu(x,dz)-\int_{R^d}(|z|^2\wedge1)\nu_n(x,dz)\right|\mu(dx)\to0 $$ as $n\to\infty$ ??
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