that $\bigcup_{j=1}^{\infty}\{i_{P,j}\}=\{i:\,(dP/d\mu)(\bar{U}_{i})>0\}$ and $\{T_{i_{P,j}}((dP/d\mu)(\bar{U}_{i_{P,j}}))^{-1}\}_{j\in\mathbb{N}}$ is sorted in ascending order with arbitrary tie-breaking (a tie occurs with probability 0). For $j\in\mathbb{N},\,u\in\mathcal{U}$, define the *mapped Poisson process with respect to $P$* as $$\left\{ \tilde{U}_{P}\left(\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}},\,j\right),\,\tilde{T}_{P}\left(\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}},\,j\right)\right\} _{j\in\mathbb{N}},\label{eq:mappedpp}$$ where $$\tilde{T}_{P}\left(\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}},\,j\right):=T_{i_{P,j}}\left(\frac{dP}{d\mu}(\bar{U}_{i_{P,j}})\right)^{-1},$$ $$\tilde{U}_{P}\left(\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}},\,j\right):=\bar{U}_{i_{P,j}}.$$ For $P,Q\ll\mu$ probability measures over $\mathcal{U}$, define $i_{P,1},i_{P,2},\ldots\in\mathbb{N}$ and $i_{Q,1},i_{Q,2},\ldots\in\mathbb{N}$ as above. Define $$\Upsilon_{P\Vert Q}\left(\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}},\,j\right):=\min\{k\in\mathbb{N}:\,i_{Q,k}=i_{P,j}\},$$ where the minimum is $\infty$ if such $k$ does not exist. We omit $\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}}$ and only write $\tilde{U}_{P}(j)$, $\tilde{T}_{P}(j)$, $\Upsilon_{P\Vert Q}(j)$ if the Poisson process is clear from the context. Note that, with probability 1, we have either $\tilde{U}_{Q}(\Upsilon_{P\Vert Q}(j))=\tilde{U}_{P}(j)$ or $\Upsilon_{P\Vert Q}(j)=\infty$. Also, for any $j,k\in\mathbb{N}$, $\Upsilon_{P\Vert Q}(j)=k\Leftrightarrow\Upsilon_{Q\Vert P}(k)=j$. Loosely speaking, $\Upsilon_{P\Vert Q}(j)$ can be regarded as “$\tilde{U}_{Q}^{-1}(\tilde{U}_{P}(j))$” (if there are no atoms in $\mu$), i.e., finding the $j$-th point in the mapped Poisson process w.r.t. $P$, then finding its index in the mapped Poisson process w.r.t. $Q$.

While $dP/d\mu$ is only uniquely defined up to a $\mu$-null set, changing the value of $dP/d\mu$ on a $\mu$-null set will only affect the values of $\{\tilde{U}_{P}(j),\,\tilde{T}_{P}(j)\}_{j\in\mathbb{N}}$ on a null set with respect to the distribution of $\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}}$, since the probability that there exists $\bar{U}_{i}$ in that $\mu$-null set is zero. Therefore $\{\tilde{U}_{P}(j),\,\tilde{T}_{P}(j)\}_{j\in\mathbb{N}}$ is uniquely defined up to a null 