\cap A \cap A_{s_0}) / ((1-p)\mu(A)) \notag\\
    &= \int_0^\infty  u^{-2} 
    \sP\Bigl(u V^{(1)}(t_0) > s_0,\, 
    \big(u V^{(1)}(t_1),\dots, u V^{(1)}(t_k)\big) \leq 
    \mathbf{s} u V^{(1)}(t_0) \Bigr) \rd u\notag\\
    &=  
    \int_0^\infty  \int_{[s_0 u^{-1},\, \infty)}  u^{-2} 
      \sP\Bigl(V^{(1)}(t_0)\in \rd y 
      \Big|\Delta\mathbf V^{(1)} \leq \mathbf{s} \Bigr)
       \sP(\Delta\mathbf V^{(1)} \leq \mathbf{s} )\rd u
      \notag\\    
    &=  
    \int_{[{\mathbf 0}, \mathbf s]} \int_{[0,\infty)} y s_0^{-1} \cdot 
      \sP\Bigl(V^{(1)}(t_0)\in \rd y \Big| 
      \Delta\mathbf V^{(1)}=\mathbf z \Bigr)
      \sP(\Delta\mathbf V^{(1)}\in \rd{\mathbf z})
      \notag\\
      &=  
      s_0^{-1}\int_{[{\mathbf 0}, \mathbf s]} \sE\Bigl( V^{(1)}(t_0) \Big| 
    \Delta\mathbf V^{(1)}=\mathbf z \Bigr)
    \sP(\Delta\mathbf V^{(1)}\in \rd{\mathbf z}).
    \label{mu_expl}
  \end{aligned}$$ Equation shows that the convergence in holds for all continuity points ${\mathbf s}\in [0, \infty)^{k}$ of the distribution function of $\Delta{\mathbf V}^{(1)}$. Since $s_0\geq 1$ was arbitrary, this concludes the proof.

1.  If $V^{(1)}(t_0)$ is stochastically independent of the increments $\Delta\mathbf V^{(1)}$, we simply have $\sP(\Delta\mathbf{\tilde{V}}^{(1)}\in d \mathbf z) =
      \sP(\Delta\mathbf{{V}}^{(1)}\in d \mathbf z)$.

2.  If $p=\sP(V(t_0) = 0)=0$, the exponent measure $\mu$ of any finite-dimensional vector $\Xi=(\xi(t_0), \ldots, \xi(t_k))$, $t_0,
      \ldots, t_k\in T$, $k\in{\mathbb N}$, satisfies the condition $\mu\left( \{0\}\times [0,\infty)^k \right)=0,$ and following Proposition \[calculateW\], the incremental representation of $\Xi$ according to is given by $\Xi = \max_{i\in{\mathbb N}}
        U_i \cdot (1, \Delta\mathbf{\tilde{V}}_i)^\top$, where $\Delta\mathbf{\tilde{V}}_i$, $i\in{\mathbb N}$, are independent copies of $\Delta\mathbf{\tilde{V}}=\Delta\mathbf{\tilde{V}}^{(1)}$.

