Let $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),\mu)$ be a $\sigma$-finite Borel measure on $d$-dimensional Euclidean space. Can one always construct a sequence of finite equivalent measures $\left\{\mu_n\right\}_{n \in \mathbb{N}}$ on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ such that $$ \lim_{n \rightarrow \infty} \frac{d\mu_n}{d\mu} \uparrow 1. $$
Motivation: This is true for both the Lebesgue measure and the counting measure on $\mathbb{R}^d$ by taking $\frac{d\mu_n}{d\mu}(x)\triangleq \begin{cases} e^{-q_n\min\{\|x\|,\frac1{\|x\|}\}}&:\, x\neq 0\\ 1 &: \, x =0 \end{cases} $, where $q_n$ runs over $\mathbb{Q}\cap(0,\infty)$.