The irrationality measure $\mu(x)$ of a real number $x$ is defined to be the supremum of the set of real numbers $\mu$ such that the inequalities $$0 < \left| x - \frac{p}{q} \right| < \frac{1}{q^\mu} \qquad (1)$$ hold for an infinite number of integer pairs $(p, q)$ with $q > 0$. Wikipedia says that $\mu(x)$ measures "how 'closely' $x$ can be approximated by rationals," but I'm very unclear about exactly how it does it, because the "approximability" of a real number seems to depend non-monotonically on $\mu$, with real numbers with low and high values of $\mu(x)$ easily approximable by rationals, and real numbers with intermediate values of $\mu(x)$ difficult to approximate by rationals.
Specifically, we have $\mu(x) \geq 1$, with
the preimage of $\mu(x) = 1$ is exactly the rationals $\mathbb{Q}$.
the preimage of $\mu(x) = 2$ contains all of the irrational algebraic numbers $\bar{Q} \setminus Q$ (by Roth's theorem), as well as almost all of the transcendental numbers (in the Lebesgue-measure sense), including $e$ and $\varphi$.
the preimage of $\mu(x) \in (2, \infty)$ is a measure-zero subset of the transcendental numbers
the preimage of $\mu(x) = \infty$ is the set of Liouville numbers (this set is "large" in the sense of having the cardinality of the continuum and being dense in the reals, but "small" in the sense of having Lebesgue measure zero).
The name "irrationality measure" seems to imply that if $\mu(x) > \mu(y)$, then $x$ is "more irrational" than $y$, i.e. is harder to approximate by a sequence of rational numbers. But in fact the opposite is true; the Louiville numbers, which have $\mu(x) = \infty$, are unusually easy to approximate by a sequence of rationals, although of course not as easy as the rationals themselves, which have $\mu(x) = 1$. How do I understand this strange non-monotonicity?
As I understand it, the problem stems entirely from the first inequality in (1), which seems extremely arbitrary and conceptually unnatural. If we remove that inequality, then the second inequality has a very nice interpretation: the error in the Diophantine approximation sequence decreases with $q$ as a power law with exponent $\mu$, and higher $\mu$ means that the error decays faster. So under this proposed modification, $\mu$ would be interpreted as a rationality measure: almost all irrational numbers would have the minimal value $\mu(x) = 2$, but a few numbers would be unusally easy to approximate and have $\mu(x) > 2$. For a rational number we would trivially have $\mu = \infty$ (under this modified definition), because the errors would vanish identically after some finite $q$. Liouville numbers would be unusual in that their Diophantine approximations would vanish with $q$ faster than any power law, although never hitting zero, so they would also have $\mu(x) = \infty$ just like the rationals.
Is there some motivation for the first inequality that I'm missing? It seems to enormously decrease the conceptual clarity of $\mu$ by making it a non-monotonic measure of Diophantine approximability.
