What is the definition of the thermodynamic limit of a thermodynamic quantity?

Statistical mechanics is all about taking thermodynamic limits and, as far as I know, there are more than one way to define such limits. Consider the following theorem:

Theorem: In the thermodynamic limit, the pressure:
$$psi(beta,h) := lim_{Lambda uparrow mathbb{Z}^{d}}psi_{Lambda}^{#}(beta, h) $$
is well-defined and independent of the sequence $Lambda uparrow mathbb{Z}^{d}$ and of the type of the boundary condition $#$.

Here, I’m using the same notation and conventions from chapter 3 of Velenik and Friedli’s book. The notation $Lambda uparrow mathbb{Z}^{d}$ stands for the convergence in the sense of Van Hove.

Definition [Convergence in the sense of Van Hove] A sequence ${Lambda_{n}}_{nin mathbb{N}}$ of (finite) subsets of $mathbb{Z}^{d}$ is said to converge to $mathbb{Z}^{d}$ in the sense of Van Hove if all three properties listed below are satisfied:

(1) ${Lambda_{n}}_{nin mathbb{N}}$ is an increasing sequence of subsets.

(2) $bigcup_{nin mathbb{N}}Lambda_{n} = mathbb{Z}^{d}$

(3) $lim_{nto infty}frac{|partial^{in}Lambda_{n}|}{|Lambda_{n}|} = 0$, where $|X|$ denotes the cardinality of the set $X$ and $partial^{in}Lambda:={iin Lambda: hspace{0.1cm} exists j inLambda^{c} hspace{0.1cm} mbox{with} hspace{0.1cm} |i-j|=1 }$

My point here is the following. Convergence in the sense of Van Hove is a notion of convergence of sets, not functions of sets. But what does $lim_{Lambdauparrow mathbb{Z}^{d}}psi^{#}_{Lambda}(beta, h)$ mean?