uniformly bounded vertex degree. If the sequence weakly converge to a random rooted simplicial complex $\mu$, then the $\ell^2$-Betti numbers of $(\mu_n)_{n\in{{\mathbb N}}}$ converge to the $\ell^2$-Betti numbers of $\mu$.

Before we start with the proof we recall the following lemmas. The first one is a consequence of the approximation theorem of Weierstraß and the second one is known as *Portmanteau theorem*:

\[lemma-compact support implies weak convergence\] Let $\mu$ be a Borel measure on ${{\mathbb R}}$ and let $(\mu_n)_{n\in{{\mathbb N}}}$ be a sequence of measures. Assume there is a compact set $C$ which contains the support of $\mu_n$ for every $n\in{{\mathbb N}}$ and assume further that $$\lim_{n\to\infty}\int_\mathbb{R}f(t)d\mu_n(t)=\int_\mathbb{R}f(t)d\mu(t)$$ holds for all polynomials $f\in \mathbb{R}[x]$. Then $\mu_n$ weakly converges to $\mu$.

\[lemma- portmanteau\] Let $\nu_n$ be a sequence of finite Borel measures on $\mathbb{R}$ which weakly converge to $\nu$. Then

1.  $\liminf_{n\to\infty} \nu_n(U)\ge \nu(U)$ for all open sets $U\subset\mathbb{R}$,

2.  $\limsup_{n\to\infty} \nu_n(A)\le \nu(A)$ for all closed sets $A\subset \mathbb{R}$.

The following lemma is the first step of the proof of the Theorem:

\[lemma-weak convergence\] For a weakly convergent sequence $\mu_n\to \mu$ of unimodular [*rrsc*]{} with $\deg(\mu_n)\le D<\infty$ for all $n\in{{\mathbb N}}$, the corresponding spectral measures $\nu^{\mu_n}_p\to \nu^\mu_p$ weakly converge.

To relax the 