The Gauss map and the sequence of continued-fraction digits

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Problem 115 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection II.1.3 "Constructions of Measures"

Define a measure μ\mu on [0,1][0,1] by the formula

μ([α,β))=log⁡21+β1+α. \mu \left([\alpha ,\beta )\right)=\log _2\frac{1+\beta }{1+\alpha }.

Prove that this measure is preserved under the transformation x↦{1/x}x\mapsto \left\{ 1/x\right\}, where {⋅}\left\{ \cdot \right\} denotes the fractional part of a number.

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Problem 116 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection II.1.3 "Constructions of Measures"

Each real number x∈[0,1]x\in [0,1] can be expanded in a continued fraction

x=1n1+1n2+⋯ x=\cfrac 1{n_1+\cfrac 1{n_2+\cdots }}

(the finite fractions correspond to the rational numbers, and the infinite ones to the irrational numbers).

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(a)

Prove that in terms of the sequence {nk}\left\{ n_k\right\} the transformation in Problem 115 has the form {nk}↦{nk+1}\left\{ n_k\right\} \mapsto \left\{ n_{k+1}\right\}.

(b)

References

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.