I am mostly looking for references to read about questions of the following kind: Let $\Omega$ be some countable/uncountable set, and let $\{A_i\}_{i=0}^\infty$ and $\{B_i\}_{i=0}^\infty$ be monotonically increasing sequences of subsets of $\Omega$. That is, for every $i \geq 0$, $A_i \subseteq \Omega$, $B_i \subseteq \Omega$, $A_i \subseteq A_{i+1}$ and $B_i \subseteq B_{i+1}$. Also suppose that the limits of these two sequences exist, and the limits coincide. How does one quantify the rate of convergence of these two sequences?
As an example, suppose, $\Omega = \mathbb{N}$, be the set of natural numbers. Say, $A_i = \{j \in \mathbb{N} | j \leq i\}$, while $B_i = \{j \in \mathbb{N} | j \leq 2\times i\}$. Does it make sense to say that the sequence $\{B_i\}_{i=0}^\infty$ converges twice as fast as the sequence $\{A_i\}_{i=0}^\infty$? How about the sequence $\{C_i\}_{i=0}^\infty$ such that $C_i = \mathbb{N}$ for all $i\geq 0$. Is there a metric on which $\{C_i\}_{i=0}^\infty$ converges faster than both $\{A_i\}_{i=0}^\infty$ and $\{B_i\}_{i=0}^\infty$. How about the sequence $\{D_i\}_{i=0}^\infty$ such that $D_0 = \varnothing$ and $D_i = \mathbb{N}$ for all $i \geq 1$?
I am more interested to understand these questions when $\Omega = \Sigma^*$ (set of all sequences over the finite alphabet $\Sigma$). In this context, a simple example will be the following. Let $\Sigma = \{0, 1\}$ and $L = \{w | w \text{ contains odd number of } 1 \text{s}\}$ be a regular language. For an $i$, define $A_i = \{ w \in L | |w| \leq i\}$ and $B_i = L \cap \{w | \text{ binary representation of } w \text{ is } i\}$. Do these sequences converge equally fast?