Understanding constructing a 'bad' probability theory example (Williams, 1991)

I am working through the book 'Probability with Martingales' by David Williams. There is a section I would like to check my understanding on. The passage is (in motivating the use of measure in probability theory):

Consider for a moment what is in some ways a bad attempt to construct a 'probability theory'. Let $K$ be the class of subsets $C$ of $\mathbb{N}$ for which the 'density'

$\rho(C) := \lim_{n \rightarrow \infty} \#\{k : 1 \leq k \leq n; \ k \in C \}$

exists. Let $C_n := \{1, 2, ..., n\}$. Then $C_n \in K$ and $C_n \rightarrow \mathbb{N}$ in the sense that $C_n \subseteq C_{n+1}, \ \forall n$ and also $\bigcup C_n = \mathbb{N}$. However, $\rho(C_n) = 0, \forall n$ but $\rho(\mathbb{N}) = 1$.

Questions

After writing this question, I saw this thread indicating that there is probably an error in $\rho(C)$ definition, and it should including the normaliser, as in $f(C)$ definition below. As such I will assume this and deleted my related questions. I still have two questions unanswered in the other thread.

1. $\rho(C_n) = 0, \forall n$. Is this just because of the general properly that the probability of a specific value in a continuous distribution is $0$? (rather than something particular to this setup).

2. Later, he says 'Let $K$ be the class of subsets $C$ of $\mathbb{N}$ for which the 'density'$f(C) = \lim_{m\rightarrow\infty} m^{-1} \#\{k: 1 \leq k \leq m; k \in C\}$ exists. We might like to think of this density (if it exists) as 'the probability that a number chosen at random belongs to $C$'. But there are many reasons why this does not conform to properly probability theory. For example, you should find elements $F$ and $G$ in $K$ for which $F \bigcap G \not\in K$'.

   In this case, do they just mean two possible subsets of $\mathbb{N}$ (of which $f(C)$ is defined) for which the intersection is not a subset of $\mathbb{N}$? i.e. this is largely unrelated to the density definition? e.g. the intersection of $F = \{1,2,3\}$, $G = \{4,5,6\}$ have intersection $\varnothing$ and so is not a subset of $\mathbb{N}$ i.e. not in $K$.

Thanks for your time.