Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Computability of the Radon-Nikodym derivative Mathieu Hoyrup, Cristóbal Rojas and Klaus Weihrauch LORIA, INRIA Nancy - France
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem • Let λ be the uniform measure over [ 0 , 1 ]. • Let f ∈ L 1 ( λ ) be nonnegative. � • Let µ ( A ) = A f d λ . • One has µ ≪ λ , i.e. for all A , λ ( A ) = 0 = ⇒ µ ( A ) = 0 .
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem • Let λ be the uniform measure over [ 0 , 1 ] . • Let f ∈ L 1 ( λ ) be nonnegative. � • Let µ ( A ) = A f d λ . • One has µ ≪ λ , i.e. for all A , λ ( A ) = 0 = ⇒ µ ( A ) = 0 . Conversely, Theorem (Radon-Nikodym, 1930) For every measure µ ≪ λ there exists f ∈ L 1 ( λ ) such that � µ ( A ) = f d λ for all Borel sets A . A f is the Radon-Nikodym derivative of µ w.r.t. λ, denoted f = d µ d λ .
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem • Let λ be the uniform measure over [ 0 , 1 ] . • Let f ∈ L 1 ( λ ) be nonnegative. � • Let µ ( A ) = A f d λ . • One has µ ≪ λ , i.e. for all A , λ ( A ) = 0 = ⇒ µ ( A ) = 0 . Conversely, Theorem (Radon-Nikodym, 1930) For every measure µ ≪ λ there exists f ∈ L 1 ( λ ) such that � µ ( A ) = f d λ for all Borel sets A . A f is the Radon-Nikodym derivative of µ w.r.t. λ, denoted f = d µ d λ . Our problem Is d µ d λ computable from µ ?
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Theorem On [ 0 , 1 ] , there is a computable measure µ ≪ λ (even µ ≤ 2 λ ) such that d µ d λ is not L 1 ( λ ) -computable.
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Theorem On [ 0 , 1 ] , there is a computable measure µ ≪ λ (even µ ≤ 2 λ ) such that d µ d λ is not L 1 ( λ ) -computable. Proof. The measure will be defined as µ ( A ) = λ ( A | K ) = λ ( A ∩ K ) where λ ( K ) K ⊆ [ 0 , 1 ] : • is a recursive compact set, • λ ( K ) > 0 is not computable (only upper semi-computable, or right-c.e.). I ǫ I 0 I 1 I 00 I 01 I 10 I 11 K
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Theorem On [ 0 , 1 ] , there is a computable measure µ ≪ λ (even µ ≤ 2 λ ) such that d µ d λ is not L 1 ( λ ) -computable. Proof cont’d. There is a computable homeomorphim φ : { 0 , 1 } N → K and µ is the push-forward φ ∗ λ of the uniform on Cantor space, so it is computable. d µ 1 λ ( K ) 1 K is not L 1 ( λ ) -computable. d λ = I ǫ I 0 I 1 I 00 I 01 I 10 I 11 K
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem (Non-)computability of RN Theorem There exists a computable measure µ ≪ λ such that d µ d λ is not L 1 ( λ ) -computable. Question How much is the Radon-Nikodym theorem non-computable?
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem (Non-)computability of RN Theorem There exists a computable measure µ ≪ λ such that d µ d λ is not L 1 ( λ ) -computable. Question How much is the Radon-Nikodym theorem non-computable? Answer No more than the Fréchet-Riesz representation theorem. Theorem RN ≤ W FR .
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem (Non-)computability of RN Theorem There exists a computable measure µ ≪ λ such that d µ d λ is not L 1 ( λ ) -computable. Question How much is the Radon-Nikodym theorem non-computable? Answer No more than the Fréchet-Riesz representation theorem. Theorem RN ≤ W FR . And even, Theorem RN ≡ W FR ≡ W EC .
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Weihrauch degrees • ≤ W is due to Weihrauch (1992).
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Weihrauch degrees • ≤ W is due to Weihrauch (1992). • According to Klaus Weihrauch, W in ≤ W stands for Wadge .
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Weihrauch degrees • ≤ W is due to Weihrauch (1992). • According to Klaus Weihrauch, W in ≤ W stands for Wadge . • Nevertheless, ≤ W is now called Weihrauch-reducibility .
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Weihrauch degrees • ≤ W is due to Weihrauch (1992). • According to Klaus Weihrauch, W in ≤ W stands for Wadge . • Nevertheless, ≤ W is now called Weihrauch-reducibility . • f ≤ W g if given x , one can compute f ( x ) applying g once.
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Weihrauch degrees • ≤ W is due to Weihrauch (1992). • According to Klaus Weihrauch, W in ≤ W stands for Wadge . • Nevertheless, ≤ W is now called Weihrauch-reducibility . • f ≤ W g if given x , one can compute f ( x ) applying g once. • f ≡ W g if f ≤ W g and g ≤ W f .
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Consider two representations En and Cf of 2 N : { n ∈ N : 100 n 1 is a subword of p } , En ( p ) = Cf ( p ) = { n ∈ N : p n = 1 } . Let E ⊆ N : • E is r.e. ⇐ ⇒ it is En -computable, • E is recursive ⇐ ⇒ it is Cf -computable. Let EC : ( 2 N , En ) → ( 2 N , Cf ) be the identity: it is not computable for these representations.
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Properties of EC • ∆ 0 2 objects can be computed from one application of EC (subsets of N , real numbers, real fonctions, points of computable metric spaces, etc.) • Actually, for f : N N → N N , f ∈ ∆ 0 2 ⇐ ⇒ f ≤ W EC . • Let J ( X ) be the Turing jump of X ⊆ N : J ≡ W EC . • EC ≡ W lim R . Due to Brattka, Gherardi, Yoshikawa, Marcone (2005-2011).
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Properties of EC Used to classify mathematical theorems: to a theorem ∀ X ∃ YP ( X , Y ) is associated the operator X �→ Y (possibly multi-valued). Due to Brattka, Gherardi, Yoshikawa, Marcone (2005-2011).
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Properties of EC Used to classify mathematical theorems: to a theorem ∀ X ∃ YP ( X , Y ) is associated the operator X �→ Y (possibly multi-valued). • Let FR be the operator associated to the Fréchet-Riesz representation theorem (on suitable spaces): EC ≡ W FR . Due to Brattka, Gherardi, Yoshikawa, Marcone (2005-2011).
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem Properties of EC Used to classify mathematical theorems: to a theorem ∀ X ∃ YP ( X , Y ) is associated the operator X �→ Y (possibly multi-valued). • Let FR be the operator associated to the Fréchet-Riesz representation theorem (on suitable spaces): EC ≡ W FR . • Let BW R be the Bolzano-Weierstrass operator: EC < W BW R . Due to Brattka, Gherardi, Yoshikawa, Marcone (2005-2011).
� � � � � Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem EC ≤ W RN Let RN be the Radon-Nikodym operator, that maps µ ≪ λ to d µ d λ ∈ L 1 ( λ ) . Corollary EC ≤ W RN . Proof. Given an enumeration of E ⊆ N : 1 construct K E such that λ ( K E ) = � ∈ E 2 − n , n / 2 apply RN to compute λ ( K E ) , 3 compute E from λ ( K E ) . � µ E enumeration of E K E EC RN d µ E λ ( K E ) characteristic function of E d λ
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem A classical proof of the Radon-Nikodym theorem works as follows: • apply the Fréchet-Riesz representation theorem to the continuous linear operator φ µ : L 2 ( λ + µ ) → R � f �→ f d µ. It gives g ∈ L 2 ( λ + µ ) such that for all f ∈ L 2 ( λ + µ ) , φ µ ( f ) = � f , g � , � � i.e. f d µ = fg d( λ + µ ) . 1 − g has the required properties for being d µ g • show that d λ .
� � � � Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem To compute the Radon-Nikodym derivative, µ φ µ RN FR d µ g 1 − g ∈ L 1 ( λ ) d λ = g ∈ L 2 ( λ + µ ) one shows that from g ∈ L 2 ( λ + µ ) one can compute g ∈ L 1 ( λ ) , knowing � that g d λ = 1 . (a simple proof can be obtained using Martin-Löf randomness!)
Radon-Nikodym theorem A counter-example Weihrauch degrees A computable Fréchet-Riesz representation theorem It was proved by Brattka and Yoshikawa that on suitable spaces, FR ≡ W EC . Hence we get EC ≤ W RN ≤ W FR ≡ W EC . FR : Fréchet-Riesz RN : Radon-Nikodym EC : Enumeration �→ Characteristic function
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