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



Approximation of $\sigma$-finite Borel measures by equivalent finite measures
Since $\mu$ is $\sigma$-finite there is a $\mu$-integrable function $f$ with $0<f\le 1$. For an increasing sequence $A\_n$ with $\bigcup\_{n\in\mathbb N} A\_n=\mathbb R^d$ define densities $f\_n=I\_{A\_n}+fI\_{A\_n^c}$. Then $\mu\_n=f\_n\cdot \mu$ are equivalent finite measures whose densities converge to $1$.


