MATH- 62052/72052 Functions of Real Variables 2. Lecture 22

MATH- 62052/72052 Functions of Real Variables 2. Lecture 22.

Artem Zvavitch

Department of Mathematical Sciences, Kent State University

Spring, 2020 This Year

Hillel Furstenberg Gregory Margulis

”for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.."

The Abel Prize 2020

"The Abel Prize was established on 1 January 2002. The purpose is to award the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 7,5 million Norwegian Krone (about 715,000 US dollars) and was awarded for the first time on 3 June 2003. The prize is awarded by the Norwegian Academy of Science and Letters, which has appointed an Abel Committee consisting of five mathematicians to review the nominated candidates and submit a recommendation for a worthy Abel laureate."

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 The Abel Prize 2020

This Year

Hillel Furstenberg Gregory Margulis

”for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.."

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 In June 2017, Analysis group at Kent State have organized NSF supported conference "Ergodic Methods in the Theory of Fractals", during which Professor Furstenberg delivered a series of ten lectures. As an outcome of this lecture series a book titled by "Ergodic Theory and Fractal Geometry" by Hillel Furstenberg was co-published by AMS and CBMS.

Hillel Furstenberg

Hillel (Harry) Furstenberg (born in Germany, in 1935. In 1939, Shortly after Kristallnacht, his family escaped to the United States) is an American-Israeli mathematician and professor emeritus at the Hebrew University of Jerusalem. He is a member of the Israel Academy of Sciences and Humanities and U.S. National Academy of Sciences and a laureate of the Abel Prize and the Wolf Prize in Mathematics. He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Hillel Furstenberg

In June 2017, Analysis group at Kent State have organized NSF supported conference "Ergodic Methods in the Theory of Fractals", during which Professor Furstenberg delivered a series of ten lectures. As an outcome of this lecture series a book titled by "Ergodic Theory and Fractal Geometry" by Hillel Furstenberg was co-published by AMS and CBMS.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Measure-preserving transformation: A mapping τ : X → X such that µ(τ −1E) = µ(E) for all E ∈ M, here τ −1(E) is a pre-image of E, i.e. τ −1(E) = {x ∈ X : τ(x) ∈ E}.

Also if τ is measure-preserving transformation + bijection + τ −1 is measure-preserving transformation, then τ is measure-preserving isomorphism.

Very basic fact τ - measure preserving, f - measurable, then f (τ(x)) is measurable and Z Z f (τ(x))dµ(x) = f (x)dµ(x). X X

This follows from {x ∈ X : f (τ(x)) > t} = τ −1({y ∈ X : f (y) > t}) and thus is measurable. −1 Also χE (τ(x)) = χτ−1(E)(x) (indeed, τ(x) ∈ E iﬀ x ∈ τ (E)). Thus the above equality for integrals is true for characteristic functions of measurable sets: Z Z Z −1 χE (τ(x))dµ(x) = χτ−1(E)(x)dµ(x) = µ(τ (E)) = µ(E) = χE (x)dµ(x) X X X and from there for simple functions and further for non-negative and measurable functions. Note that we also proved: µ({x ∈ X : f (τ(x)) > t}) = µ({y ∈ X : f (y) > t}).

Ergodic Theory: measure preserving transformations

We will play in σ-ﬁnite measure space (X, M, µ). One of our heroes will be

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Also if τ is measure-preserving transformation + bijection + τ −1 is measure-preserving transformation, then τ is measure-preserving isomorphism.

Very basic fact τ - measure preserving, f - measurable, then f (τ(x)) is measurable and Z Z f (τ(x))dµ(x) = f (x)dµ(x). X X

Ergodic Theory: measure preserving transformations

We will play in σ-ﬁnite measure space (X, M, µ). One of our heroes will be

Measure-preserving transformation: A mapping τ : X → X such that µ(τ −1E) = µ(E) for all E ∈ M, here τ −1(E) is a pre-image of E, i.e. τ −1(E) = {x ∈ X : τ(x) ∈ E}.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Very basic fact τ - measure preserving, f - measurable, then f (τ(x)) is measurable and Z Z f (τ(x))dµ(x) = f (x)dµ(x). X X

Ergodic Theory: measure preserving transformations

We will play in σ-ﬁnite measure space (X, M, µ). One of our heroes will be

Measure-preserving transformation: A mapping τ : X → X such that µ(τ −1E) = µ(E) for all E ∈ M, here τ −1(E) is a pre-image of E, i.e. τ −1(E) = {x ∈ X : τ(x) ∈ E}.

Also if τ is measure-preserving transformation + bijection + τ −1 is measure-preserving transformation, then τ is measure-preserving isomorphism.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 −1 Also χE (τ(x)) = χτ−1(E)(x) (indeed, τ(x) ∈ E iﬀ x ∈ τ (E)). Thus the above equality for integrals is true for characteristic functions of measurable sets: Z Z Z −1 χE (τ(x))dµ(x) = χτ−1(E)(x)dµ(x) = µ(τ (E)) = µ(E) = χE (x)dµ(x) X X X and from there for simple functions and further for non-negative and measurable functions. Note that we also proved: µ({x ∈ X : f (τ(x)) > t}) = µ({y ∈ X : f (y) > t}).

Ergodic Theory: measure preserving transformations

We will play in σ-ﬁnite measure space (X, M, µ). One of our heroes will be

Measure-preserving transformation: A mapping τ : X → X such that µ(τ −1E) = µ(E) for all E ∈ M, here τ −1(E) is a pre-image of E, i.e. τ −1(E) = {x ∈ X : τ(x) ∈ E}.

Also if τ is measure-preserving transformation + bijection + τ −1 is measure-preserving transformation, then τ is measure-preserving isomorphism.

Very basic fact τ - measure preserving, f - measurable, then f (τ(x)) is measurable and Z Z f (τ(x))dµ(x) = f (x)dµ(x). X X

This follows from {x ∈ X : f (τ(x)) > t} = τ −1({y ∈ X : f (y) > t}) and thus is measurable.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 (indeed, τ(x) ∈ E iﬀ x ∈ τ −1(E)). Thus the above equality for integrals is true for characteristic functions of measurable sets: Z Z Z −1 χE (τ(x))dµ(x) = χτ−1(E)(x)dµ(x) = µ(τ (E)) = µ(E) = χE (x)dµ(x) X X X and from there for simple functions and further for non-negative and measurable functions. Note that we also proved: µ({x ∈ X : f (τ(x)) > t}) = µ({y ∈ X : f (y) > t}).

Ergodic Theory: measure preserving transformations

We will play in σ-ﬁnite measure space (X, M, µ). One of our heroes will be

Measure-preserving transformation: A mapping τ : X → X such that µ(τ −1E) = µ(E) for all E ∈ M, here τ −1(E) is a pre-image of E, i.e. τ −1(E) = {x ∈ X : τ(x) ∈ E}.

Also if τ is measure-preserving transformation + bijection + τ −1 is measure-preserving transformation, then τ is measure-preserving isomorphism.

Very basic fact τ - measure preserving, f - measurable, then f (τ(x)) is measurable and Z Z f (τ(x))dµ(x) = f (x)dµ(x). X X

This follows from {x ∈ X : f (τ(x)) > t} = τ −1({y ∈ X : f (y) > t}) and thus is measurable.

Also χE (τ(x)) = χτ−1(E)(x)

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Thus the above equality for integrals is true for characteristic functions of measurable sets: Z Z Z −1 χE (τ(x))dµ(x) = χτ−1(E)(x)dµ(x) = µ(τ (E)) = µ(E) = χE (x)dµ(x) X X X and from there for simple functions and further for non-negative and measurable functions. Note that we also proved: µ({x ∈ X : f (τ(x)) > t}) = µ({y ∈ X : f (y) > t}).

Ergodic Theory: measure preserving transformations

We will play in σ-ﬁnite measure space (X, M, µ). One of our heroes will be

Measure-preserving transformation: A mapping τ : X → X such that µ(τ −1E) = µ(E) for all E ∈ M, here τ −1(E) is a pre-image of E, i.e. τ −1(E) = {x ∈ X : τ(x) ∈ E}.

Also if τ is measure-preserving transformation + bijection + τ −1 is measure-preserving transformation, then τ is measure-preserving isomorphism.

Very basic fact τ - measure preserving, f - measurable, then f (τ(x)) is measurable and Z Z f (τ(x))dµ(x) = f (x)dµ(x). X X

This follows from {x ∈ X : f (τ(x)) > t} = τ −1({y ∈ X : f (y) > t}) and thus is measurable. −1 Also χE (τ(x)) = χτ−1(E)(x) (indeed, τ(x) ∈ E iﬀ x ∈ τ (E)).

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Note that we also proved: µ({x ∈ X : f (τ(x)) > t}) = µ({y ∈ X : f (y) > t}).

Ergodic Theory: measure preserving transformations

We will play in σ-ﬁnite measure space (X, M, µ). One of our heroes will be

Measure-preserving transformation: A mapping τ : X → X such that µ(τ −1E) = µ(E) for all E ∈ M, here τ −1(E) is a pre-image of E, i.e. τ −1(E) = {x ∈ X : τ(x) ∈ E}.

Also if τ is measure-preserving transformation + bijection + τ −1 is measure-preserving transformation, then τ is measure-preserving isomorphism.

Very basic fact τ - measure preserving, f - measurable, then f (τ(x)) is measurable and Z Z f (τ(x))dµ(x) = f (x)dµ(x). X X

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Ergodic Theory: measure preserving transformations

We will play in σ-ﬁnite measure space (X, M, µ). One of our heroes will be

Measure-preserving transformation: A mapping τ : X → X such that µ(τ −1E) = µ(E) for all E ∈ M, here τ −1(E) is a pre-image of E, i.e. τ −1(E) = {x ∈ X : τ(x) ∈ E}.

Also if τ is measure-preserving transformation + bijection + τ −1 is measure-preserving transformation, then τ is measure-preserving isomorphism.

Very basic fact τ - measure preserving, f - measurable, then f (τ(x)) is measurable and Z Z f (τ(x))dµ(x) = f (x)dµ(x). X X

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83. What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

but this is NOT what we really want. The issue is #E may not be equal to τ −1(E), indeed if E = {1} then τ −1(E) = ∅. This particular τ and E also help us to see that our integral formula would not work if we ask for #τ(E) = #E (and not for #τ −1(E) = #E). We want Z Z f (τ(x))dµ(x) = f (x)dµ(x). Z Z

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83. What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 or τ(n) = n − 83. What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Notice that in this case

#τ(E) = #E,

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83. What about τ(n) = 2n?

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 but this is NOT what we really want. The issue is #E may not be equal to τ −1(E), indeed if E = {1} then τ −1(E) = ∅. This particular τ and E also help us to see that our integral formula would not work if we ask for #τ(E) = #E (and not for #τ −1(E) = #E). We want Z Z f (τ(x))dµ(x) = f (x)dµ(x). Z Z

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83. What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 The issue is #E may not be equal to τ −1(E), indeed if E = {1} then τ −1(E) = ∅. This particular τ and E also help us to see that our integral formula would not work if we ask for #τ(E) = #E (and not for #τ −1(E) = #E). We want Z Z f (τ(x))dµ(x) = f (x)dµ(x). Z Z

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83. What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

but this is NOT what we really want.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 This particular τ and E also help us to see that our integral formula would not work if we ask for #τ(E) = #E (and not for #τ −1(E) = #E). We want Z Z f (τ(x))dµ(x) = f (x)dµ(x). Z Z

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83. What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

but this is NOT what we really want. The issue is #E may not be equal to τ −1(E), indeed if E = {1} then τ −1(E) = ∅.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 We want Z Z f (τ(x))dµ(x) = f (x)dµ(x). Z Z

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83. What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83. What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Measure preserving transformations: Examples

Take X = Z and µ - counting measure, i.e. µ(E) = #E = the number of elements in E.

An example of a measure preserving transformation τ can be a unit transformation τ(n) = n + 1. Actually τ is a measure preserving isomorphism.

Clearly in above example we could also take τ(n) = n + 27 or τ(n) = n − 83. What about τ(n) = 2n? Notice that in this case

#τ(E) = #E,

Plug f = χE , E = {1} and τ(n) = 2n: Z Z χE (τ(n))dµ(n) = µ{n ∈ Z : τ(z) ∈ {1}} = 0 6= 1 = µ(E) = χE (n)dµ(n). Z Z

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Another example would be a transformation τ(x) = Ax where A is a d by d matrix with determinant 1 or −1.

T = R/Z, i.e. we identify all real numbers mod 1 and create a unit circle. µ is a a measure induced from the Lebesgue measure on R, i.e. T = (0,1] and µ is m restricted to (0,1]. Fix α ∈ R and let τ(x) = x + α mod 1. The transformation is well deﬁned on T and measure preserving (please, check! It may help to see it just a rotation of our circle by 2πα).

n n We can generalize the above example to the torus T = (0,1] with Lebesgue measure on its Borel σ-algebra, and, as before consider addition modulo 1 in each coordinate:

τ(x1,...,xn) = (x1 + a1,...,xn + an),

n where a = (a1,...,an) ∈ T .

Measure preserving transformations: Examples

d Take X = R and µ = m standard Lebesgue measure and τ(x) = x + h for some ﬁxed d h ∈ R . Then τ is a measure preserving isomorphism.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 T = R/Z, i.e. we identify all real numbers mod 1 and create a unit circle. µ is a a measure induced from the Lebesgue measure on R, i.e. T = (0,1] and µ is m restricted to (0,1]. Fix α ∈ R and let τ(x) = x + α mod 1. The transformation is well deﬁned on T and measure preserving (please, check! It may help to see it just a rotation of our circle by 2πα).

n n We can generalize the above example to the torus T = (0,1] with Lebesgue measure on its Borel σ-algebra, and, as before consider addition modulo 1 in each coordinate:

τ(x1,...,xn) = (x1 + a1,...,xn + an),

n where a = (a1,...,an) ∈ T .

Measure preserving transformations: Examples

d Take X = R and µ = m standard Lebesgue measure and τ(x) = x + h for some ﬁxed d h ∈ R . Then τ is a measure preserving isomorphism. Another example would be a transformation τ(x) = Ax where A is a d by d matrix with determinant 1 or −1.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 µ is a a measure induced from the Lebesgue measure on R, i.e. T = (0,1] and µ is m restricted to (0,1]. Fix α ∈ R and let τ(x) = x + α mod 1. The transformation is well deﬁned on T and measure preserving (please, check! It may help to see it just a rotation of our circle by 2πα).

n n We can generalize the above example to the torus T = (0,1] with Lebesgue measure on its Borel σ-algebra, and, as before consider addition modulo 1 in each coordinate:

τ(x1,...,xn) = (x1 + a1,...,xn + an),

n where a = (a1,...,an) ∈ T .

Measure preserving transformations: Examples

T = R/Z, i.e. we identify all real numbers mod 1 and create a unit circle.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Fix α ∈ R and let τ(x) = x + α mod 1. The transformation is well deﬁned on T and measure preserving (please, check! It may help to see it just a rotation of our circle by 2πα).

n n We can generalize the above example to the torus T = (0,1] with Lebesgue measure on its Borel σ-algebra, and, as before consider addition modulo 1 in each coordinate:

τ(x1,...,xn) = (x1 + a1,...,xn + an),

n where a = (a1,...,an) ∈ T .

Measure preserving transformations: Examples

T = R/Z, i.e. we identify all real numbers mod 1 and create a unit circle. µ is a a measure induced from the Lebesgue measure on R, i.e. T = (0,1] and µ is m restricted to (0,1].

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 n n We can generalize the above example to the torus T = (0,1] with Lebesgue measure on its Borel σ-algebra, and, as before consider addition modulo 1 in each coordinate:

τ(x1,...,xn) = (x1 + a1,...,xn + an),

n where a = (a1,...,an) ∈ T .

Measure preserving transformations: Examples

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Measure preserving transformations: Examples

n n We can generalize the above example to the torus T = (0,1] with Lebesgue measure on its Borel σ-algebra, and, as before consider addition modulo 1 in each coordinate:

τ(x1,...,xn) = (x1 + a1,...,xn + an),

n where a = (a1,...,an) ∈ T .

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 We say that τ is measurable if τ −1(E) is measurable for every E ∈ M. Note that in this case we can deﬁne a new −1 + −1 measure µ(τ ): M → R ∪ {+∞} acting on E ∈ M simply by µ(τ (E)) (check that µ(τ −1) is a measure!) and is called push-forward of µ with respect to τ. Thus τ is measure preserving transformation iﬀ µ(τ −1) = µ on M.

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

Proof: Assume we have that µ(τ −1(A)) = µ(A), for all A ∈ A, then µ = µ(τ −1) on A, but then we can use the uniqueness part of extension theorem from algebra A to σ-algebra B to declare that µ = µ(τ −1) on B and thus τ is measure preserving! The converse is trivial, clearly if µ is measure preserving on B is is measure preserving on a subset of B (and thus on A).

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Consider a transformation τ acting on (X,M,µ).

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Note that in this case we can deﬁne a new −1 + −1 measure µ(τ ): M → R ∪ {+∞} acting on E ∈ M simply by µ(τ (E)) (check that µ(τ −1) is a measure!) and is called push-forward of µ with respect to τ. Thus τ is measure preserving transformation iﬀ µ(τ −1) = µ on M.

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Consider a transformation τ acting on (X,M,µ). We say that τ is measurable if τ −1(E) is measurable for every E ∈ M.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Thus τ is measure preserving transformation iﬀ µ(τ −1) = µ on M.

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Consider a transformation τ acting on (X,M,µ). We say that τ is measurable if τ −1(E) is measurable for every E ∈ M. Note that in this case we can deﬁne a new −1 + −1 measure µ(τ ): M → R ∪ {+∞} acting on E ∈ M simply by µ(τ (E)) (check that µ(τ −1) is a measure!) and is called push-forward of µ with respect to τ.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Proof: Assume we have that µ(τ −1(A)) = µ(A), for all A ∈ A, then µ = µ(τ −1) on A, but then we can use the uniqueness part of extension theorem from algebra A to σ-algebra B to declare that µ = µ(τ −1) on B and thus τ is measure preserving! The converse is trivial, clearly if µ is measure preserving on B is is measure preserving on a subset of B (and thus on A).

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 then µ = µ(τ −1) on A, but then we can use the uniqueness part of extension theorem from algebra A to σ-algebra B to declare that µ = µ(τ −1) on B and thus τ is measure preserving! The converse is trivial, clearly if µ is measure preserving on B is is measure preserving on a subset of B (and thus on A).

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

Proof: Assume we have that µ(τ −1(A)) = µ(A), for all A ∈ A,

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 but then we can use the uniqueness part of extension theorem from algebra A to σ-algebra B to declare that µ = µ(τ −1) on B and thus τ is measure preserving! The converse is trivial, clearly if µ is measure preserving on B is is measure preserving on a subset of B (and thus on A).

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

Proof: Assume we have that µ(τ −1(A)) = µ(A), for all A ∈ A, then µ = µ(τ −1) on A,

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 The converse is trivial, clearly if µ is measure preserving on B is is measure preserving on a subset of B (and thus on A).

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

A useful fact

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 A useful fact

Let B be a σ-algebra generated by an algebra A, then µ is preserved by a measurable transformation τ if and only if

µ(τ −1(A)) = µ(A), for all A ∈ A,

i.e., it is enough to check the measure preserving relation for the elements on the generating algebra A and then it automatically holds for all elements of B.

This facts makes checking three previous examples in particular easy, indeed, it is enough to check that a measure of a segment (or box) is preserved.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 −1 x x+1 Consider x ∈ (0,1] then τ (x) = { 2 , 2 } and for interval [a,b] ∈ T (i.e. a,b ∈ (0,1]), we get a b a + 1 b + 1 τ −1([a,b]) = [ , ] ∪ [ , ]. 2 2 2 2 Notice that b ≤ a + 1 so

µ(τ −1([a,b])) = (b − a)/2 + [(b + 1) − (a + 1)]/2 = b − a = µ([a,b]).

From here it is easy to continue the argument to the union of segments and use the extension theorem (in short a fact on the previous slide) from Algebra to σ-algebra. A small side note: Please, note again that µ(τ(0,1/2]) = µ((0,1]) = 1 6= µ((0,1/2]) which again explains the use of τ −1.

Measure preserving transformations: Examples

A bit less trivial example on T T = R/Z, i.e. we identify all real numbers mod 1 and create a unit circle. µ is a a measure induced from the Lebesgue measure on R, i.e. T = (0,1] and µ is m restricted to (0,1]. Let τ(x) = 2x mod 1.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 and for interval [a,b] ∈ T (i.e. a,b ∈ (0,1]), we get a b a + 1 b + 1 τ −1([a,b]) = [ , ] ∪ [ , ]. 2 2 2 2 Notice that b ≤ a + 1 so

µ(τ −1([a,b])) = (b − a)/2 + [(b + 1) − (a + 1)]/2 = b − a = µ([a,b]).

Measure preserving transformations: Examples

−1 x x+1 Consider x ∈ (0,1] then τ (x) = { 2 , 2 }

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Notice that b ≤ a + 1 so

µ(τ −1([a,b])) = (b − a)/2 + [(b + 1) − (a + 1)]/2 = b − a = µ([a,b]).

Measure preserving transformations: Examples

−1 x x+1 Consider x ∈ (0,1] then τ (x) = { 2 , 2 } and for interval [a,b] ∈ T (i.e. a,b ∈ (0,1]), we get a b a + 1 b + 1 τ −1([a,b]) = [ , ] ∪ [ , ]. 2 2 2 2

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 From here it is easy to continue the argument to the union of segments and use the extension theorem (in short a fact on the previous slide) from Algebra to σ-algebra. A small side note: Please, note again that µ(τ(0,1/2]) = µ((0,1]) = 1 6= µ((0,1/2]) which again explains the use of τ −1.

Measure preserving transformations: Examples

−1 x x+1 Consider x ∈ (0,1] then τ (x) = { 2 , 2 } and for interval [a,b] ∈ T (i.e. a,b ∈ (0,1]), we get a b a + 1 b + 1 τ −1([a,b]) = [ , ] ∪ [ , ]. 2 2 2 2 Notice that b ≤ a + 1 so

µ(τ −1([a,b])) = (b − a)/2 + [(b + 1) − (a + 1)]/2 = b − a = µ([a,b]).

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 A small side note: Please, note again that µ(τ(0,1/2]) = µ((0,1]) = 1 6= µ((0,1/2]) which again explains the use of τ −1.

Measure preserving transformations: Examples

−1 x x+1 Consider x ∈ (0,1] then τ (x) = { 2 , 2 } and for interval [a,b] ∈ T (i.e. a,b ∈ (0,1]), we get a b a + 1 b + 1 τ −1([a,b]) = [ , ] ∪ [ , ]. 2 2 2 2 Notice that b ≤ a + 1 so

µ(τ −1([a,b])) = (b − a)/2 + [(b + 1) − (a + 1)]/2 = b − a = µ([a,b]).

From here it is easy to continue the argument to the union of segments and use the extension theorem (in short a fact on the previous slide) from Algebra to σ-algebra.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Measure preserving transformations: Examples

−1 x x+1 Consider x ∈ (0,1] then τ (x) = { 2 , 2 } and for interval [a,b] ∈ T (i.e. a,b ∈ (0,1]), we get a b a + 1 b + 1 τ −1([a,b]) = [ , ] ∪ [ , ]. 2 2 2 2 Notice that b ≤ a + 1 so

µ(τ −1([a,b])) = (b − a)/2 + [(b + 1) − (a + 1)]/2 = b − a = µ([a,b]).

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Let G : X → X be the Gauss map: n0 x = 0 G(x) = 1 h x i x ∈ (0, 1] where hai is a fractional part of a (i.e. a subtracted it’s integer value).

1 The Gauss measure µ is the measure deﬁned by the density (1+x) log 2 , i.e. Z 1 µ(E) = dx, (1 + x) log 2 E for all E ∈ M. A strange log 2 factor is easy to explain, consider [a, b] ⊂ [0, 1]: Z b 1 1 b 1 1 + b µ([a, b]) = dx = log(1 + x) = log (1 + x) log 2 log 2 a log 2 1 + a a In particular, µ([0, 1]) = 1 and thus µ is a probability measure on [0, 1].

The Gauss map is a measure preserving transformation on (X, M, µ).

Yet, another example: Gauss Map

Let X = [0, 1] and M be standard Borel σ- algebra on X.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 1 The Gauss measure µ is the measure deﬁned by the density (1+x) log 2 , i.e. Z 1 µ(E) = dx, (1 + x) log 2 E for all E ∈ M. A strange log 2 factor is easy to explain, consider [a, b] ⊂ [0, 1]: Z b 1 1 b 1 1 + b µ([a, b]) = dx = log(1 + x) = log (1 + x) log 2 log 2 a log 2 1 + a a In particular, µ([0, 1]) = 1 and thus µ is a probability measure on [0, 1].

The Gauss map is a measure preserving transformation on (X, M, µ).

Yet, another example: Gauss Map

Let X = [0, 1] and M be standard Borel σ- algebra on X. Let G : X → X be the Gauss map: n0 x = 0 G(x) = 1 h x i x ∈ (0, 1] where hai is a fractional part of a (i.e. a subtracted it’s integer value).

The Gauss map is a measure preserving transformation on (X, M, µ).

Yet, another example: Gauss Map

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 A strange log 2 factor is easy to explain, consider [a, b] ⊂ [0, 1]: Z b 1 1 b 1 1 + b µ([a, b]) = dx = log(1 + x) = log (1 + x) log 2 log 2 a log 2 1 + a a In particular, µ([0, 1]) = 1 and thus µ is a probability measure on [0, 1].

The Gauss map is a measure preserving transformation on (X, M, µ).

Yet, another example: Gauss Map

1 The Gauss measure µ is the measure deﬁned by the density (1+x) log 2 , i.e. Z 1 µ(E) = dx, (1 + x) log 2 E for all E ∈ M.

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 In particular, µ([0, 1]) = 1 and thus µ is a probability measure on [0, 1].

The Gauss map is a measure preserving transformation on (X, M, µ).

Yet, another example: Gauss Map

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Yet, another example: Gauss Map

The Gauss map is a measure preserving transformation on (X, M, µ).

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 1 1 1 1 1 Let Gn be a part of G restricted to n+1 , n ; Gn(x) = G(x) = x − n for x ∈ n+1 , n . −1 Gn is surjective and monotone, which will help us to compute G ([a, b]) for [a, b] ⊂ [0, 1]. −1 −1 Indeed, G ([a, b]) consists of aunion countably many disjoint intervals of the form Gn ([a, b]):

−1 n 1 o h 1 1 i G ([a, b]) = {x : Gn(x) ∈ [a, b]} = x ∈ [0, 1]: a ≤ − n ≤ b = , . n x b + n a + n −1 ∞ Now we are left with direct computation. Using that {Gn ([a, b])}n=1 are disjoint we get X X h 1 1 i µ(G−1([a, b])) = µ(∪G−1([a, b])) = µ(G−1([a, b])) = µ , . n n b + n a + n

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

n0 x = 0 G(x) = 1 h x i x ∈ (0, 1] where hai is a fractional part of a (i.e. a subtracted it’s integer value).

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 −1 Gn is surjective and monotone, which will help us to compute G ([a, b]) for [a, b] ⊂ [0, 1]. −1 −1 Indeed, G ([a, b]) consists of aunion countably many disjoint intervals of the form Gn ([a, b]):

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

n0 x = 0 G(x) = 1 h x i x ∈ (0, 1] where hai is a fractional part of a (i.e. a subtracted it’s integer value).

1 1 1 1 1 Let Gn be a part of G restricted to n+1 , n ; Gn(x) = G(x) = x − n for x ∈ n+1 , n .

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 which will help us to compute G−1([a, b]) for [a, b] ⊂ [0, 1]. −1 −1 Indeed, G ([a, b]) consists of aunion countably many disjoint intervals of the form Gn ([a, b]):

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

n0 x = 0 G(x) = 1 h x i x ∈ (0, 1] where hai is a fractional part of a (i.e. a subtracted it’s integer value).

1 1 1 1 1 Let Gn be a part of G restricted to n+1 , n ; Gn(x) = G(x) = x − n for x ∈ n+1 , n .

Gn is surjective and monotone,

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 −1 −1 Indeed, G ([a, b]) consists of aunion countably many disjoint intervals of the form Gn ([a, b]):

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

n0 x = 0 G(x) = 1 h x i x ∈ (0, 1] where hai is a fractional part of a (i.e. a subtracted it’s integer value).

1 1 1 1 1 Let Gn be a part of G restricted to n+1 , n ; Gn(x) = G(x) = x − n for x ∈ n+1 , n . −1 Gn is surjective and monotone, which will help us to compute G ([a, b]) for [a, b] ⊂ [0, 1].

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 −1 n 1 o h 1 1 i G ([a, b]) = {x : Gn(x) ∈ [a, b]} = x ∈ [0, 1]: a ≤ − n ≤ b = , . n x b + n a + n −1 ∞ Now we are left with direct computation. Using that {Gn ([a, b])}n=1 are disjoint we get X X h 1 1 i µ(G−1([a, b])) = µ(∪G−1([a, b])) = µ(G−1([a, b])) = µ , . n n b + n a + n

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

n0 x = 0 G(x) = 1 h x i x ∈ (0, 1] where hai is a fractional part of a (i.e. a subtracted it’s integer value).

1 1 1 1 1 Let Gn be a part of G restricted to n+1 , n ; Gn(x) = G(x) = x − n for x ∈ n+1 , n . −1 Gn is surjective and monotone, which will help us to compute G ([a, b]) for [a, b] ⊂ [0, 1]. −1 −1 Indeed, G ([a, b]) consists of aunion countably many disjoint intervals of the form Gn ([a, b]):

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 −1 ∞ Now we are left with direct computation. Using that {Gn ([a, b])}n=1 are disjoint we get X X h 1 1 i µ(G−1([a, b])) = µ(∪G−1([a, b])) = µ(G−1([a, b])) = µ , . n n b + n a + n

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

n0 x = 0 G(x) = 1 h x i x ∈ (0, 1] where hai is a fractional part of a (i.e. a subtracted it’s integer value).

−1 n 1 o h 1 1 i G ([a, b]) = {x : Gn(x) ∈ [a, b]} = x ∈ [0, 1]: a ≤ − n ≤ b = , . n x b + n a + n

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

n0 x = 0 G(x) = 1 h x i x ∈ (0, 1] where hai is a fractional part of a (i.e. a subtracted it’s integer value).

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Z 1/(a+n) X dx X 1 1/(a+n) = = log(1 + x) (1 + x) log 2 log 2 1/(b+n) 1/(b+n) 1 Xh 1 + a + n 1 + b + n i = log − log log 2 a + n b + n Using tha the above sums are telescoping we get

N N X 1 + a + n X log = [log(1 + a + n) − log(a + n)] = log(1 + a + N) − log(a + 1) a + n n=1 n=1 so 1 µ(G−1([a, b])) = lim [log(1 + a + N) − log(a + 1) − log(1 + b + N) + log(b + 1)] log 2 N→∞ 1 h 1 + a + N b + 1 i 1 h b + 1 i = lim log + log = log = µ([a, b]). log 2 N→∞ 1 + b + N a + 1 log 2 a + 1 So we proved that G is measure preserving on intervals, thus applying extension fact from above we get that G is a measure preserving transformation.

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

X X h 1 1 i µ(G−1([a, b])) = µ(∪G−1([a, b])) = µ(G−1([a, b])) = µ , = n n b + n a + n

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 1 Xh 1 + a + n 1 + b + n i = log − log log 2 a + n b + n Using tha the above sums are telescoping we get

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

X X h 1 1 i µ(G−1([a, b])) = µ(∪G−1([a, b])) = µ(G−1([a, b])) = µ , = n n b + n a + n Z 1/(a+n) X dx X 1 1/(a+n) = = log(1 + x) (1 + x) log 2 log 2 1/(b+n) 1/(b+n)

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Using tha the above sums are telescoping we get

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

X X h 1 1 i µ(G−1([a, b])) = µ(∪G−1([a, b])) = µ(G−1([a, b])) = µ , = n n b + n a + n Z 1/(a+n) X dx X 1 1/(a+n) = = log(1 + x) (1 + x) log 2 log 2 1/(b+n) 1/(b+n) 1 Xh 1 + a + n 1 + b + n i = log − log log 2 a + n b + n

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 so 1 µ(G−1([a, b])) = lim [log(1 + a + N) − log(a + 1) − log(1 + b + N) + log(b + 1)] log 2 N→∞ 1 h 1 + a + N b + 1 i 1 h b + 1 i = lim log + log = log = µ([a, b]). log 2 N→∞ 1 + b + N a + 1 log 2 a + 1 So we proved that G is measure preserving on intervals, thus applying extension fact from above we get that G is a measure preserving transformation.

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

X X h 1 1 i µ(G−1([a, b])) = µ(∪G−1([a, b])) = µ(G−1([a, b])) = µ , = n n b + n a + n Z 1/(a+n) X dx X 1 1/(a+n) = = log(1 + x) (1 + x) log 2 log 2 1/(b+n) 1/(b+n) 1 Xh 1 + a + n 1 + b + n i = log − log log 2 a + n b + n Using tha the above sums are telescoping we get

N N X 1 + a + n X log = [log(1 + a + n) − log(a + n)] = log(1 + a + N) − log(a + 1) a + n n=1 n=1

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 1 h 1 + a + N b + 1 i 1 h b + 1 i = lim log + log = log = µ([a, b]). log 2 N→∞ 1 + b + N a + 1 log 2 a + 1 So we proved that G is measure preserving on intervals, thus applying extension fact from above we get that G is a measure preserving transformation.

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

N N X 1 + a + n X log = [log(1 + a + n) − log(a + n)] = log(1 + a + N) − log(a + 1) a + n n=1 n=1 so 1 µ(G−1([a, b])) = lim [log(1 + a + N) − log(a + 1) − log(1 + b + N) + log(b + 1)] log 2 N→∞

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 So we proved that G is measure preserving on intervals, thus applying extension fact from above we get that G is a measure preserving transformation.

Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2 Gauss Map and measure on [0,1]

The Gauss map is a measure preserving transformation on (X, M, µ).

Artem Zvavitch MATH-6/72052 Functions of Real Variables 2