ii) follows from symmetry and multilinearity of the intersection product ([Ful98, Proposition 2.5]). For iii) we reduce first to the case where $\mathfrak{X}^{\prime}$ and $\mathfrak{X}$ have reduced special fibre. Let $\mathfrak{Y}^{\prime}$ respectively $\mathfrak{Y}$ be the canonical formal models with reduced special fibre as in 4. This construction is functorial and we obtain a commutative diagram

Assuming that we know the claim for reduced special fibres we obtain

	$\displaystyle\Deg(\varphi^{\textup{an}})c_{1}(\mathfrak{L}_{1})\wedge...\wedge c_{1}(\mathfrak{L}_{n})$	$\displaystyle=\Deg(\varphi^{\prime an})(\iota^{\textup{an}})_{\ast}(c_{1}(\iota^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\iota^{\ast}\mathfrak{L}_{n}))$
		$\displaystyle=(\iota^{\textup{an}})_{\ast}(\varphi^{\prime an})_{\ast}(c_{1}(\varphi^{\prime\ast}\iota^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\varphi^{\prime\ast}\iota^{\ast}\mathfrak{L}_{n}))$
		$\displaystyle=(\varphi^{\textup{an}})_{\ast}(\iota^{\prime an})_{\ast}(c_{1}(\iota^{\prime\ast}\varphi^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\iota^{\prime\ast}\varphi^{\ast}\mathfrak{L}_{n}))$
		$\displaystyle=(\varphi^{\textup{an}})_{\ast}(c_{1}(\varphi^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\varphi^{\ast}\mathfrak{L}_{n})).$

So from now on assume that $\mathfrak{X}^{\prime}$ and $\mathfrak{X}$ have reduced special fibre. Let $Y$ be an irreducible component of $\tilde{\mathfrak{X}}$ with corresponding Shilov point $\zeta_{Y}$. Let $\zeta_{1},...,\zeta_{r}$ be the preimages of $\zeta_{Y}$ under $\varphi^{\textup{an}}$ with corresponding irreducible components $Y_{1},...,Y_{r}$ of $\tilde{\mathfrak{X}^{\prime}}$. If $Y$ is proper then clearly all the $Y_{i}$ are proper. If on the other hand one of the $Y_{i}$ is proper then $Y$ is proper by [GW10, Proposition 12.59]. In this case we can use the projection formula to calculate:

	$\displaystyle(\varphi^{\textup{an}})_{\ast}\left(c_{1}(\varphi^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\varphi^{\ast}\mathfrak{L}_{n})\right)(\zeta_{Y})$	$\displaystyle=\sum_{i=1}^{r}c_{1}(\varphi^{\ast}\mathfrak{L}_{1})\wedge...\wedge c_{1}(\varphi^{\ast}\mathfrak{L}_{n})(\zeta_{i})$
		$\displaystyle=\sum_{i=1}^{r}\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(\tilde{\varphi}_{\ast}Y_{i})$
		$\displaystyle=\sum_{i=1}^{r}\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(Y)\cdot[\tilde{K}(Y_{i}):\tilde{K}(Y)]$

As already mentioned in Definition 4.2, $\varphi^{\textup{an}}$ is finite outside a lower dimensional analytic subset. Hence we may apply equation (3) in the proof of [Gub98, Proposition 4.5] to see that the last term in the display equals $\Deg(\varphi^{\textup{an}})\cdot c_{1}(\mathfrak{L}_{1})\wedge...\wedge c_{1}(\mathfrak{L}_{n})(\zeta_{Y})$.
On the other hand, if $Y$ is an irreducible component of $\tilde{\mathfrak{X}}^{\prime}$ whose image is not an irreducible component of $\tilde{\mathfrak{X}}$ then its degree with respect to the line bundles $\varphi^{\ast}\mathfrak{L}_{1},...,\varphi^{\ast}\mathfrak{L}_{n}$ is $0$ by the projection formula, as the image is of lower dimension. This proves iii).
For i) let $\mathfrak{X}^{\textup{an}}=\sum_{j}m_{j}X_{j}$ be the decomposition into prime cycles. It is then enough to prove the claim for each $X_{j}$ and by definition of the measure we may hence assume that $\mathfrak{X}$ has irreducible and reduced generic fibre and reduced special fibre. Let $S$ be the set of all $\zeta_{Y}$ where $Y$ is a proper irreducible component of $\tilde{\mathfrak{X}}$ with $\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(Y)\neq 0$. Then $S$ is discrete as $\red^{-1}(Y)$ is an open neighbourhood of $\zeta_{Y}$ which does not contain any other points of $S$. Furthermore $X$ is the union of all $\red^{-1}(Y)$ where $Y$ runs over all irreducible components of $\tilde{\mathfrak{X}}$ and as all of these sets contain at most one point of $S$ and by paracompactness of $X$, every $x\notin S$ has an open neighbourhood which does not intersect $S$ and hence $S$ is closed. By definition $c_{1}(\mathfrak{L}_{1})\wedge...\wedge c_{1}(\mathfrak{L}_{n})$ is of the desired form and its support is contained in the relative interior of $X$ over $K$ by Corollary A.4. ∎

Let $\mathfrak{X}^{\textup{an}}=\sum_{j}m_{j}X_{j}$ be the decomposition into prime cycles. It is enough to prove the claim for the closures $\overline{X}_{j}$ of $X_{j}$ in $\mathfrak{X}$. We may hence assume that $\mathfrak{X}^{\textup{an}}$ is irreducible and reduced. Furthermore by passing to a dominating model as in 4, we may assume that the special fibre $\tilde{\mathfrak{X}}$ of $\mathfrak{X}$ is reduced. As $\mathfrak{X}^{\textup{an}}$ has no boundary, every irreducible component of $\tilde{\mathfrak{X}}$ is proper by Corollary A.4 and hence using commutativity of the intersection product ([Gub98, Theorem 5.9]) we obtain

	$\displaystyle\int_{\mathfrak{X}^{\textup{an}}}f_{0}\;c_{1}(\mathfrak{L}_{1})\wedge...\wedge c_{1}(\mathfrak{L}_{n})$	$\displaystyle=\sum_{\begin{subarray}{c}Y\in\irr(\tilde{\mathfrak{X}})\end{subarray}}f_{0}(\zeta_{Y})\cdot\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(Y)$
		$\displaystyle=\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}\left(\cyc(\Div_{\mathfrak{L}_{0}}(1))\right)$
		$\displaystyle=\Deg_{\mathfrak{L}_{0},\mathfrak{L}_{2},...,\mathfrak{L}_{n}}\left(\cyc(\Div_{\mathfrak{L}_{1}}(1))\right)$
		$\displaystyle=\sum_{\begin{subarray}{c}Y\in\irr(\tilde{\mathfrak{X}})\end{subarray}}f_{1}(\zeta_{Y})\cdot\Deg_{\mathfrak{L}_{0},\mathfrak{L}_{2},...,\mathfrak{L}_{n}}(Y)$
		$\displaystyle=\int_{\mathfrak{X}^{\textup{an}}}f_{1}\;c_{1}(\mathfrak{L}_{0})\wedge c_{1}(\mathfrak{L}_{2})\wedge...\wedge c_{1}(\mathfrak{L}_{n}).$

∎

Let $W_{2}=\sum_{j}m_{j}X_{j}$ be the decomposition of $W_{2}$ into prime cycles and for each $j$ let $(X_{ij})_{i\in I_{j}}$ be the irreducible components of $W_{1}$ with $X_{ij}\subseteq X_{j}\cap W_{1}$. Then $W_{1}=\sum_{j,i}m_{j}X_{ij}$ is the decomposition of $W_{1}$ into prime cycles. Furthermore, the intersection of any two irreducible components of $W_{1}$ does not contain a Shilov point as it is of lower dimension and hence does not meet the support of the measures of interest. By linearity in the irreducible components we may therefore assume that $W_{1}$ and $W_{2}$ are irreducible and reduced. Let $\mathfrak{X}_{2}$ be a formal model of $W_{2}$ with reduced special fibre on which there exist formal models of $\overline{L}_{1},...,\overline{L}_{n}$. Let $\mathfrak{X}_{1}$ be a formal model of $W_{1}$ which exists by paracompactness of $W_{1}$, see Remark 3.2. After possibly blowing up, the inclusion $W_{1}\hookrightarrow W_{2}$ induces a morphism $\iota:\mathfrak{X}_{1}\rightarrow\mathfrak{X}_{2}$ ([Bos14, Theorem 8.4.3]). Let $x\in\overset{\circ}{W_{1}}$. As both measures are discrete it is enough to show that they have the same mass at $x$. Let $\Int(W_{i})$ denote the relative interior of $W_{i}$ over $K$ in the sense of [Ber93, 1.5]. If $x\in\Int(W_{2})$ then $x\in\Int(W_{1})$ by [Ber93, Proposition 1.5.5 (ii)]. Conversely if $x\in\Int(W_{1})$ then there exists an affinoid neighbourhood $V$ of $x$ in $W_{1}$ such that $x$ is in the relative interior of $V$ over $K$. But $V$ is also a neighbourhood of $x$ in $W_{2}$ as $x\in\overset{\circ}{W_{1}}$ and therefore $x\in\Int(W_{2})$. Hence $x\in\Int(W_{1})$ if and only if $x\in\Int(W_{2})$. If this is not the case then by definition of the measures and Corollary A.4, both of them are zero at $x$. So assume that $x\in\Int(W_{1})$. Choose a locally finite cover $(\mathfrak{U}_{i})_{i\in I}$ of $\mathfrak{X}_{1}$ by open affine formal subschemes and let $\mathfrak{U}$ be the union of all $\mathfrak{U}_{i}$ which contain $\red(x)$. Then $\mathfrak{U}$ is an open and quasi-compact formal subscheme of $\mathfrak{X}_{1}$. Analogously choose a cover $(\mathfrak{V}_{i})_{i\in J}$ of $\mathfrak{X}_{2}$ by open affine formal subschemes. As $\iota(\mathfrak{U})$ is quasi-compact, there is a finite subcover of it. Let $\mathfrak{V}$ be the union of the sets in this subcover and add all $\mathfrak{V}_{i}$ with $\red(x)\in\mathfrak{V}_{i}$. Then also $\mathfrak{V}$ is an open and quasi-compact formal subscheme of $\mathfrak{X}_{2}$ and $\iota$ induces a morphism $\mathfrak{U}\rightarrow\mathfrak{V}$. By [BL93, Corollary 5.4] there is an admissible formal blowing up $\mathfrak{V}^{\prime}\rightarrow\mathfrak{V}$ such that the induced morphism $\mathfrak{U}^{\prime}\rightarrow\mathfrak{V}^{\prime}$ is an open immersion.
Let $Y$ be an irreducible component of $\tilde{\mathfrak{X}}_{2}$ with corresponding divisorial point $\zeta_{Y}=x$. Then $Y\subseteq\tilde{\mathfrak{V}}$ by definition and hence we may calculate the mass of $c_{1}(\overline{L}_{1})\wedge...\wedge c_{1}(\overline{L}_{n})$ at $x$ using $\mathfrak{V}$. By Proposition 4.5 iii) we may also use $\mathfrak{V}^{\prime}$. So let $Y^{\prime}$ be the irreducible component of $\tilde{\mathfrak{V}}^{\prime}$ corresponding to $x$. Since $\red(x)\in\tilde{\mathfrak{U}}^{\prime}$ we see that $Y\cap\tilde{\mathfrak{U}}^{\prime}$ is an irreducible component of $\tilde{\mathfrak{U}}^{\prime}$. Additionally, by Corollary A.4, $Y^{\prime}$ and $Y^{\prime}\cap\tilde{\mathfrak{U}}^{\prime}$ are proper and hence $Y^{\prime}=Y^{\prime}\cap\tilde{\mathfrak{U}}^{\prime}$ and it is an irreducible component of $\tilde{\mathfrak{U}}^{\prime}$. It’s image in $\tilde{\mathfrak{U}}$ is a proper irreducible component of $\tilde{\mathfrak{U}}$ and hence also an irreducible component of $\tilde{\mathfrak{X}}_{1}$. By the same argumentation as above we may use $\mathfrak{U}^{\prime}$ instead of $\mathfrak{X}_{1}$ to calculate the mass of $c_{1}\left(\overline{L}_{1}\Big|_{W_{1}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n}\Big|_{W_{1}}\right)$ at $x$. This shows that the mass of the two measures is equal at $x$ in this case.
Conversely, if $Y$ is an irreducible component of $\tilde{\mathfrak{X}}_{1}$ with corresponding divisorial point $\zeta_{Y}=x$ then $Y\subseteq\tilde{\mathfrak{U}}$ by definition. Again we may use $\mathfrak{U}^{\prime}$ to calculate the mass at $x$ and we denote the corresponding irreducible component by $Y^{\prime}$. Then the closure $\overline{Y}^{\prime}$ of $Y^{\prime}$ in $\tilde{\mathfrak{V}}^{\prime}$ is an irreducible component of $\tilde{\mathfrak{V}}^{\prime}$ with corresponding divisorial point $\zeta_{\overline{Y}^{\prime}}=x$ and hence by the above $\overline{Y}^{\prime}=Y^{\prime}$. Therefore $c_{1}(\overline{L}_{1})\wedge...\wedge c_{1}(\overline{L}_{n})$ and $c_{1}\left(\overline{L}_{1}\Big|_{W_{1}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n}\Big|_{W_{1}}\right)$ coincide at $x$. ∎

By Vojta’s version of Nagata’s compactification theorem ([Voj, Theorem 5.7]) we may assume that $X$ is proper. We show by reverse induction over $m\in\{0,...,n\}$ that the claim holds when for some choice of pairwise different $i_{1},...,i_{n}\in\{1,...,n\}$ the sequences $\Big(\|\cdot\|_{i_{1},k}\Big)_{k\in\mathbb{N}},...,\Big(\|\cdot\|_{i_{m},k}\Big)_{k\in\mathbb{N}}$ are constant with respect to $k$. The case $m=n$ is clear. So let $0\leq m<n$ and assume that the claim holds for $m+1$. For $j\in\{m+1,...,n\}$ we can write $\|\cdot\|_{i_{j},k}=\|\cdot\|_{i_{j},1}\otimes\|\cdot\|^{\prime}_{j,k}$ for a sequence of piecewise $\mathbb{Q}$-linear metrics $\Big(\|\cdot\|^{\prime}_{j,k}\Big)_{k\in\mathbb{N}}$ on $\mathcal{O}_{X^{\textup{an}}}\Big|_{V}$ converging uniformly to a continuous metric $\|\cdot\|^{\prime}_{j}$ on $\mathcal{O}_{X^{\textup{an}}}\Big|_{V}$. Denote by $\overline{\mathcal{O}}_{j,k}$ the line bundle $\mathcal{O}_{X^{\textup{an}}}\Big|_{V}$ endowed with the metric $\|\cdot\|^{\prime}_{j,k}$. We show that

$$\left(\mu_{m,k}:=c_{1}(\overline{L}_{i_{1},1})\wedge...\wedge c_{1}(\overline{L}_{i_{m},1})\wedge c_{1}(\overline{L}_{i_{m+1},k})\wedge...\wedge c_{1}(\overline{L}_{i_{n},k})\right)_{k\in\mathbb{N}}$$

is a Cauchy sequence with respect to the weak topology on the space of Borel-measures on $V$. Thus we have to show that for all continuous functions $f$ on $X$ with compact support in $V$:

$$\left|\int_{V}f\;\mu_{m,k}-\int_{V}f\;\mu_{m,k^{\prime}}\right|\underset{k,k^{\prime}\rightarrow\infty}{\longrightarrow}0.$$

Let $W$ be a compact strictly $K$-analytic domain with $\supp(f)\subseteq\overset{\circ}{W}$ and $W\subseteq V$. By [GM19, Proposition 2.7] we may extend the metrics from $W$ to $X^{\textup{an}}$ and hence assume that they are defined on the whole space. Hence by Chow’s lemma and the projection formula we may assume that $X$ is projective. Then by [Gub03, Proposition 10.5] any formal model of $X$ is dominated by a projective model. Any formal line bundle on this model becomes semipositive after tensoring with $\mathcal{O}(n)$ for $n$ big enough by using Serre’s theorem ([Har77, Theorem II.5.17]) on the special fibre. As a consequence one can write any formal metric on any line bundle on $X$ as a quotient of two semipositive formal metrics (on possibly different line bundles). We will see below, that $\mu_{m,k}(Z)$ is bounded with respect to $k$ for every compact subset $Z\subseteq V$. Hence, as the set of piecewise $\mathbb{Q}$-linear metrics is dense in the space of continuous metrics on $\mathcal{O}_{X^{\textup{an}}}$ with respect to uniform convergence (Proposition 3.13), we may assume that $f=-\log\|1\|$ for a formal metric $\|\cdot\|$ on $\mathcal{O}_{X^{\textup{an}}}$. Then we can write $\|\cdot\|=\|\cdot\|_{+}/\|\cdot\|_{-}$ for two semipositive formal metrics $\|\cdot\|_{+},\|\cdot\|_{-}$ on some line bundles $L_{+}$ respectively $L_{-}$ on $X^{\textup{an}}$. In fact $L_{+}=L_{-}$ but we will use the notation $\overline{L}_{+}$ and $\overline{L}_{-}$ to distinguish between the two metrics. Write $\overline{\mathcal{O}}_{X}^{f}$ for the line bundle $\mathcal{O}_{X^{\textup{an}}}\Big|_{V}$ endowed with the metric $\|1\|=e^{-f}$ and to shorten notation $\mu_{m}:=c_{1}(\overline{L}_{i_{1},1})\wedge...\wedge c_{1}(\overline{L}_{i_{m},1})$ which is a purely formal notation. Furthermore without loss of generality assume $i_{1}=1,...,i_{m}=m$. We have

	$\displaystyle\Big|$	$\displaystyle\int_{V}f\;\mu_{m,k}-\int_{V}f\;\mu_{m,k^{\prime}}\Big|$
		$\displaystyle=\Big|\sum_{i=1}^{n-m}\int_{V}f\;\mu_{m}\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{m+i,k}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)$
		$\displaystyle\phantom{\Big|\sum_{i=1}^{n}}-\int_{V}f\;\mu_{m}\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{L}_{m+i,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)\Big|$
		$\displaystyle=\Big|\sum_{i=1}^{n-m}\int_{V}f\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{L}_{m+i,1}\otimes\overline{\mathcal{O}}_{m+i,k}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...$
		$\displaystyle\phantom{\Big|\sum_{i=1}^{n}}-\int_{V}f\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{L}_{m+i,1}\otimes\overline{\mathcal{O}}_{m+i,k^{\prime}}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...\Big|$
		$\displaystyle=\Big|\sum_{i=1}^{n-m}\int_{V}f\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{m+i,k}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...$
		$\displaystyle\phantom{\Big|\sum_{i=1}^{n}}-\int_{V}f\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{m+i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{m+i,k^{\prime}}\right)\wedge c_{1}\left(\overline{L}_{m+i+1,k^{\prime}}\right)\wedge...\Big|$

Since the support of $f$ is contained in $V$ and by Lemma 4.8 these last integrals depend only on the restrictions of the metrics to $V$. Hence we may instead consider them as integrals over $X^{\textup{an}}$ which allows us to use Lemma 4.6 as $X^{\textup{an}}$ has no boundary ([Ber90, Theorem 3.4.1]). In combination with an index shift, the last term amounts to

	$\displaystyle\Big|$	$\displaystyle\sum_{i=m+1}^{n}\int_{X^{\textup{an}}}-\log\|1\|^{\prime}_{i,k}\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...$
		$\displaystyle-\int_{X^{\textup{an}}}-\log\|1\|^{\prime}_{i,k^{\prime}}\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\Big|$

As any point in $X^{\textup{an}}\setminus\supp(f)$ has a strictly $K$-analytic neighbourhood on which $f$ vanishes, the support of $\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...$ is contained in $\supp(f)$ by Lemma 4.8. So the last display equals

	$\displaystyle\Big|$	$\displaystyle\sum_{i=m+1}^{n}\int_{\supp(f)}-\log\|1\|^{\prime}_{i,k}\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...$
		$\displaystyle-\int_{\supp(f)}-\log\|1\|^{\prime}_{i,k^{\prime}}\;\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\Big|$
		$\displaystyle=\Big|\sum_{i=m+1}^{n}\int_{\supp(f)}\log\Big(\|1\|^{\prime}_{i,k^{\prime}}/\|1\|^{\prime}_{i,k}\Big)\;...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{\mathcal{O}}_{X}^{f}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\Big|$
		$\displaystyle\leq 2\cdot\sum_{i=m+1}^{n}\sup_{x\in\supp(f)}\left|\log\Big(\|1\|^{\prime}_{i,k}(x)/\|1\|^{\prime}_{i,k^{\prime}}(x)\Big)\right|$
		$\displaystyle\phantom{\leq\sum_{i=m+}^{n}}\cdot\max_{s\in\{+,-\}}\mu_{m}\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{L}_{s}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...(\supp(f))\underset{k,k^{\prime}\rightarrow\infty}{\longrightarrow}0.$

Here the last term converges to zero as $\sup_{x\in\supp(f)}\left|\log\Big(\|1\|^{\prime}_{i,k}(x)/\|1\|^{\prime}_{i,k^{\prime}}(x)\Big)\right|$ tends to zero by uniform convergence of $\|\cdot\|^{\prime}_{i,k}$ and compactness of $\supp(f)$ and $\mu_{m}\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{L}_{s}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)$ are positive measures on $V$ which converge by the induction hypothesis weakly to a positive Radon measure which implies that their mass of $\supp(f)$ is bounded with respect to $k,k^{\prime}$. To go into more detail, let $g$ be a continuous non-negative function on $V$ with compact support such that $g(x)>1$ for all $x\in\supp(f)$. The existence of such a function follows for example from a partition of unity argument ([Flo03, 1.5.1]) applied to the open cover $\{\overline{V}\setminus\supp(f),V\}$ of the closure $\overline{V}$ of $V$ (note that $\overline{V}$ is compact as $X$ is proper over $K$). Then

	$\displaystyle\mu_{m}$	$\displaystyle\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{L}_{s}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)(\supp(f))$
		$\displaystyle\leq\int g\;\mu_{m}\wedge c_{1}\left(\overline{L}_{m+1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{i-1,k}\right)\wedge c_{1}\left(\overline{L}_{s}\right)\wedge c_{1}\left(\overline{L}_{i+1,k^{\prime}}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n,k^{\prime}}\right)$

where the last term converges for $k,k^{\prime}\rightarrow\infty$ and is hence bounded with respect to $k,k^{\prime}$.
We now define a positive linear functional on the space of continuous functions with compact support in $V$ by

	$\displaystyle C_{c}(V)$	$\displaystyle\rightarrow\mathbb{R}_{\geq 0},$
	$\displaystyle f$	$\displaystyle\mapsto\lim_{k\rightarrow\infty}\int_{V}f\;\mu_{m,k}.$

By the Riesz Representation Theorem ([Rud87, Theorem 2.14]) this corresponds to a positive Radon measure $\mu$ on $V$ and we have $\mu_{m,k}\rightarrow\mu$ weakly for $k\rightarrow\infty$.
It remains to show that $\mu_{m,k}(Z)$ is bounded with respect to $k$ for every compact subset $Z\subseteq V$. So let $Z\subseteq V$ be compact and $f$ a continuous non-negative function on $V$ with compact support such that $f(x)>1$ for all $x\in Z$. As above the existence of such a function follows from a partition of unity argument ([Flo03, 1.5.1]) applied to the open cover $\{\overline{V}\setminus Z,V\}$ of the closure $\overline{V}$ of $V$. Again we may assume that $f$ is a model function, i.e. of the from $-\log\|\cdot\|$ for a piecewise $\mathbb{Q}$-linear metric $\|\cdot\|$ on $\mathcal{O}_{X^{\textup{an}}}$ (we can even assume that $\|\cdot\|$ is a formal metric) and we use the same notation as above. To be more precise, let $\epsilon>0$ such that $f(x)>1+\epsilon$ for all $x\in Z$. First extend $f$ to $X^{\textup{an}}$ by zero and then define a new function $\tilde{f}$ by $\tilde{f}(x)=f(x)-\epsilon/2$. By Proposition 3.13 we may approximate $\tilde{f}$ by a model function $\phi$ such that $|\phi(x)-\tilde{f}(x)|<\epsilon/2$ for all $x\in X^{\textup{an}}$. Then by [GM19, Proposition 2.12 (d)], $\max\{0,\phi\}$ is a model function on $X^{\textup{an}}$ with compact support in $V$ which is greater than one at $Z$. We have

	$\displaystyle\sup_{k\in\mathbb{N}}\mu_{m,k}(Z)$	$\displaystyle\leq\sup_{k\in\mathbb{N}}\int_{V}f\;\mu_{m,k}$
		$\displaystyle=\sup_{k\in\mathbb{N}}\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{L}_{m+1,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle=\sup_{k\in\mathbb{N}}\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{L}_{m+1,1}\otimes\overline{\mathcal{O}}_{m+1,k})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle=\sup_{k\in\mathbb{N}}\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{L}_{m+1,1})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle\phantom{\lim_{k\rightarrow\infty}}+\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{\mathcal{O}}_{m+1,k})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$

Again using Lemma 4.6 and the same argumentation as above for the second summand this amounts to

	$\displaystyle\sup_{k\in\mathbb{N}}$	$\displaystyle\int_{\supp(f)}f\;\mu_{m}\wedge c_{1}(\overline{L}_{m+1,1})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle\phantom{\lim_{k\rightarrow\infty}}+\int_{\supp(f)}-\log\|1\|^{\prime}_{m+1,k}\;\mu_{m}\wedge c_{1}(\overline{\mathcal{O}}_{X}^{f})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})$
		$\displaystyle\leq\sup_{k\in\mathbb{N}}\sup_{x\in V}f(x)\cdot\mu_{m}\wedge c_{1}(\overline{L}_{m+1,1})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})(\supp(f))$
		$\displaystyle\phantom{\lim_{k\rightarrow\infty}}+\sup_{k\in\mathbb{N}}\sup_{x\in\supp(f)}\left|\log\Big(\|1\|^{\prime}_{m+1,k}(x)\Big)\right|$
		$\displaystyle\phantom{\lim_{k\rightarrow\infty}+}\cdot 2\cdot\max_{s\in\{+,-\}}\mu_{m}\wedge c_{1}(\overline{L}_{s})\wedge c_{1}(\overline{L}_{m+2,k})\wedge...\wedge c_{1}(\overline{L}_{n,k})(\supp(f))$

By the induction hypothesis all measures appearing in this last term converge for $k\rightarrow\infty$. Hence the measure of $\supp(f)$ is bounded with respect to $k$. Furthermore $\sup_{x\in V}f(x)<\infty$ as $f$ has compact support in $V$ and $\sup_{x\in\supp(f)}\left|\log\Big(\|1\|^{\prime}_{m+1,k}(x)\Big)\right|$ is bounded with respect to $k$ by uniform convergence of $\Big(\|\cdot\|^{\prime}_{m+1,k}\Big)_{k\in\mathbb{N}}$ and compactness of $\supp(f)$. We conclude that the last term is bounded with respect to $k$. This proves the induction step. The claim is then the case $m=0$. ∎