Editing Liouville number (section)

==Liouville numbers and measure ==
From the point of view of [[measure theory]], the set of all Liouville numbers <math>L</math> is small. More precisely, its [[Lebesgue measure]], <math>\lambda(L)</math>, is zero. The proof given follows some ideas by [[John C. Oxtoby]].<ref name="oxtoby">{{Cite book | last = Oxtoby | first = John C. | year = 1980 | title = Measure and Category | series = Graduate Texts in Mathematics | volume = 2 | edition = Second | publisher = Springer-Verlag | isbn = 0-387-90508-1 | location = New York-Berlin | mr=0584443 | doi=10.1007/978-1-4684-9339-9}}</ref>{{Rp|8}}

For positive integers <math>n>2</math> and <math>q\geq2</math> set:

:<math>V_{n,q}=\bigcup\limits_{p=-\infty}^\infty \left(\frac{p}{q}-\frac{1}{q^n},\frac{p}{q}+\frac{1}{q^n}\right)</math>

then

:<math>L\subseteq \bigcup_{q=2}^\infty V_{n,q}.</math>

Observe that for each positive integer <math>n\geq2</math> and <math>m\geq1</math>, then

:<math>L\cap (-m,m)\subseteq \bigcup\limits_{q=2}^\infty V_{n,q}\cap(-m,m)\subseteq \bigcup\limits_{q=2}^\infty\bigcup\limits_{p=-mq}^{mq} \left( \frac{p}{q}-\frac{1}{q^n},\frac{p}{q}+\frac{1}{q^n}\right).</math>

Since

: <math> \left|\left(\frac{p}{q}+\frac{1}{q^n}\right)-\left(\frac{p}{q}-\frac{1}{q^n}\right)\right|=\frac{2}{q^n}</math>

and <math>n>2</math> then

: <math>
\begin{align}
\mu(L\cap (-m,\, m)) & \leq\sum_{q=2}^\infty\sum_{p=-mq}^{mq}\frac{2}{q^n} = \sum_{q=2}^\infty \frac{2(2mq+1)}{q^n} \\[6pt]
& \leq (4m+1)\sum_{q=2}^\infty\frac{1}{q^{n-1}} \leq (4m+1) \int^\infty_1 \frac{dq}{q^{n-1}}\leq\frac{4m+1}{n-2}.
\end{align}
</math>

Now

:<math>\lim_{n\to\infty}\frac{4m+1}{n-2}=0</math>

and it follows that for each positive integer <math>m</math>, <math>L\cap (-m,m)</math> has Lebesgue measure zero. Consequently, so has <math>L</math>.

In contrast, the Lebesgue measure of the set of ''all'' real transcendental numbers is [[Infinity|infinite]] (since the set of algebraic numbers is a [[null set]]).

One could show even more - the set of Liouville numbers has [[Hausdorff dimension]] 0 (a property strictly stronger than having Lebesgue measure 0).