Editing Khinchin's constant (section)

==Sketch of proof==
The proof presented here was arranged by [[Czesław Ryll-Nardzewski]]<ref>{{Cite journal |last=Ryll-Nardzewski |first=C. |date=1951 |title=On the ergodic theorems (II) (Ergodic theory of continued fractions) |url=http://www.impan.pl/get/doi/10.4064/sm-12-1-74-79 |journal=Studia Mathematica |language=en |volume=12 |issue=1 |pages=74–79 |doi=10.4064/sm-12-1-74-79 |issn=0039-3223|url-access=subscription }}</ref> and is much simpler than Khinchin's original proof which did not use [[ergodic theory]].

Since the first coefficient ''a''<sub>0</sub> of the continued fraction of ''x'' plays no role in Khinchin's theorem and since the [[rational numbers]] have [[Lebesgue measure]] zero, we are reduced to the study of irrational numbers in the [[unit interval]], i.e., those in <math>I=[0,1]\setminus\mathbb{Q}</math>. These numbers are in [[bijection]] with infinite [[continued fraction]]s of the form [0;&nbsp;''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;...], which we simply write [''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;...], where ''a''<sub>1</sub>, ''a''<sub>2</sub>,&nbsp;... are [[positive integer]]s. Define a transformation ''T'':''I''&nbsp;&rarr;&nbsp;''I'' by

:<math>T([a_1,a_2,\dots])=[a_2,a_3,\dots].\,</math>

The transformation ''T'' is called the [[Gauss–Kuzmin–Wirsing operator]]. For every  [[Borel set|Borel subset]] ''E'' of ''I'', we also define the [[Gauss–Kuzmin distribution|Gauss&ndash;Kuzmin measure]] of ''E''

:<math>\mu(E)=\frac{1}{\ln 2}\int_E\frac{dx}{1+x}.</math>

Then ''&mu;'' is a [[probability measure]] on the [[Sigma-algebra|''&sigma;''-algebra]] of Borel subsets of ''I''. The measure ''&mu;'' is [[Equivalence (measure theory)|equivalent]] to the Lebesgue measure on ''I'', but it has the additional property that the transformation ''T'' [[measure-preserving transformation|preserves]] the measure ''&mu;''. Moreover, it can be proved that ''T'' is an [[ergodic transformation]] of the [[measurable space]] ''I'' endowed with the probability measure ''&mu;'' (this is the hard part of the proof). The [[ergodic theorem]] then says that for any ''&mu;''-[[integrable function]] ''f'' on ''I'', the average value of <math>f \left( T^k x \right)</math> is the same for almost all <math>x</math>:

:<math>\lim_{n\to\infty} \frac 1n\sum_{k=0}^{n-1}(f\circ T^k)(x)=\int_I f d\mu\quad\text{for }\mu\text{-almost all }x\in I.</math>

Applying this to the function defined by ''f''([''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;...]) = ln(''a''<sub>1</sub>), we obtain that

:<math>\lim_{n\to\infty}\frac 1n\sum_{k=1}^{n}\ln a_k=\int_I f \, d\mu = \sum_{r=1}^\infty\ln\left[1+\frac{1}{r(r+2)}\right]\log_2r</math>

for almost all [''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;...] in ''I'' as ''n''&nbsp;&rarr;&nbsp;&infin;.

Taking the [[exponential function|exponential]] on both sides, we obtain to the left the [[geometric mean]] of the first ''n'' coefficients of the continued fraction, and to the right Khinchin's constant.