From the given measure $\mu,$ how to construct another measure $\mu^{\ast}$; so that $d\mu^{\ast}(y)= (1+y^{2})d\mu(y)$?

Question

Put $\mu= \sum_{n\in \mathbb Z}c_{n}\delta_{n};$ where $\delta_{n}$ is the unit Dirac mass at $n.$ We note that, $\mu$ is a complex Borel measure on $\mathbb R$ and the total variation of $\mu,$ that is, $\|\mu\|= \sum_{n\in \mathbb Z} |c_{n}|.$

Now from the given $\mu$ as above, I wants to construct, another measure $\mu^{\ast}$ on $\mathbb R$ such that $d\mu^{\ast}(y)= (1+y^{2}) d\mu(y).$

Then,

My Question is: What should be the choice of $\mu^{\ast}$ and what can we about the total variation of $\mu^{\ast},$ that is, what is $\|\mu^{\ast}\|$ ?

Answer 1

Just define $\mu^*$ to have the density $y \mapsto (1+y^2)$ with respect to $\mu$, that is we have for $A \subseteq \mathbb R$: $$ \mu^*(A) = \int_{ A} (1+y^2)\, d\mu = \sum_{n \in A} c_n\cdot (1+n^2) $$ Hence $\mu^* = \sum_{n} (1+n^2)c_n\delta_n$ and the total variation is $\sum_n |c_n|(1+n^2)$.