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 $\mathbb{Q}\cap(0,\infty)$.