Minimum and maximum limit of a set sequences in Italian notation

I'm not sure whether this question has already been treated on TeX.SE. In this Italian document the minimum (maximum) limit is denoted with \lim' (\lim'').

Using this code

    \documentclass[a4paper,12pt]{article}    \usepackage{mathtools,amssymb}    \usepackage{geometry}    \DeclareMathOperator*{\minlim}{minlim}    \DeclareMathOperator*{\maxlim}{maxlim}    \begin{document}    Sia $S$ un insieme qualunque non vuoto. Detta $\{X_n\}$ una successione di sottoinsiemi di $S$. Si chiama minimo limite di $\{X_n\}$ e si denota con il simbolo $\lim'_{n\to \infty} X_n$, l'insieme dei punti di $S$ che godono della seguente proprietà: $\exists\nu\in\mathbb{N} : x\in X_n$ per $n>\nu$. Si ha ovviamente $\lim'_{n\to \infty} X_n=\cup_{n=1}^\infty(X_{n+1}\cap X_{n+2}\cap\cdots )$. Si chiama massimo limite di $\{X_n\}$ e si denota con il simbolo $\lim''_{n\to \infty} X_n$, l'insieme dei punti di $S$ che godono della seguente proprietà: $x\in X_n$ per infiniti valori dell'indice $n$. Si ha ovviamente $\lim''_{n\to \infty} X_n=\cap_{n=1}^\infty(X_{n+1}\cup X_{n+2}\cup\cdots)$.    Le notazioni    \[\varlimsup_{n} X_n, \minlim_{n} X_n, \lim'_{n\to \infty}X_n, \liminf_{n} X_n \textup{ e }\minlim_{n} X_n, \lim''_{n\to \infty} X_{n}\]    hanno lo stesso significato. Stesso dicasi per    \[\varliminf_{n} X_n, \minlim_{n} X_n, \lim{'}_{n\to \infty}X_n, \liminf_{n} X_n  \textup{ e } \maxlim_{n} X_n, \lim{''}_{n\to \infty} X_{n}, \limsup_{n} X_n\]    \end{document}

I have noticed several typesetting issues:

a. When using ' or '' as superscripts to indicate limit inferior/superior, the line spacing is increasing in text mode (I not know if is increasing also in math-mode). To avoid this, I tried wrapping the apostrophes in braces ({'} and {''}), but:

• In mathmode, an unwanted horizontal space appears between \lim' and the subscript n \to \infty.

b. Additionally:

• The vertical spacing between \liminf and its subscript differs from that of \limsup.
• The same issue between the spacing between \varliminf and the subscript _{n} and the same between \varlimsup_{n} and the subscript _{n}.

I'm looking for suggestions to improve the typographical quality of these notations, ideally with minimal changes to the existing code.

In the following image I have hopefully highlighted all the problems with connecting lines.