For a real number $x$, we define the (Liouville-Roth) irrationality measure of $x$, here denoted by $\mu(x)$, as the infimum, with respect to the poset $(\mathbb{R}_0^+ \cup \{\infty\}, \le)$, of the (possibly empty) set of all real exponents $s > 0$ such that the inequality $$0 < \left|x - \frac{m}{n}\right| < \frac{1}{n^s} $$ holds true for finitely many pairs $(m,n) \in \mathbb{Z} \times \mathbb{N}^+$ (with the convention that $\sup(\emptyset) := \infty$).

Now, fix $\tilde{x} \in \mathbb{R}$ and let $M: \mathbb{R} \cup \{\infty\} \to \mathbb{R} \cup \{\infty\}$ be the MÃ¶bius transformation $$ x \mapsto \left\{\begin{array}{ll} (ax+b)/(cx+d) & \text{if }x \ne \infty \\ a/c & \text{if }c \ne 0 \text{ and }x = \infty \\ \infty & \text{if }(c = 0 \text{ and }x = \infty) \text{ or }(c \ne 0 \text{ and }x = - d/c)\\ \end{array}\right., $$ where $a,b,c,d \in \mathbb Q$ and $ad-bc \ne 0$. It is then possible to prove that $\mu(\tilde{x}) = \mu(M(\tilde{x}))$ provided that $c\tilde{x}+d \ne 0$, and here come my questions:

Q1.Is there a "standard" reference for this? I checked Mahler'sLectures on Diophantine Approximationsand Schmidt'sDiophantine Approximations and Diophantine Equations, but the result doesn't seem to be mentioned in either of them.Q2.From the historical point of view, who was the first one who proved the result? Maybe J. Liouville, in his early work in diophantine approximation theory?

Thank you beforehand for any help.

Heights in diophantine geometryby Bombieri and Gubler, particularly Prop. 2.8.19. – Vesselin Dimitrov Sep 26 '13 at 22:55Diophantine approximation in projective space: cecm.sfu.ca/~choi/paper/metric.pdf . – Vesselin Dimitrov Sep 26 '13 at 23:07An introduction to diophantine approximation(Ch. I, Sect. 3, p. 11) for the basic case of ${\rm PGL}(2,\mathbb{Q})$ and approximations to reals by rational numbers. – Salvo Tringali Sep 26 '13 at 23:48