Computability of the Radon-Nikodym derivative

Transcription

1 Computability of the Radon-Nikodym derivative Mathieu Hoyrup, Cristóbal Rojas and Klaus Weihrauch LORIA, INRIA Nancy - France

2 Let λ be the uniform measure over [0, 1]. Let f L 1 (λ) be nonnegative. Let µ(a) = A f dλ. One has µ λ, i.e. for all A, λ(a) = 0 = µ(a) = 0.

3 Let λ be the uniform measure over [0, 1]. Let f L 1 (λ) be nonnegative. Let µ(a) = A f dλ. 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 dλ for all Borel sets A. A f is the Radon-Nikodym derivative of µ w.r.t. λ, denoted f = dµ dλ.

4 Let λ be the uniform measure over [0, 1]. Let f L 1 (λ) be nonnegative. Let µ(a) = A f dλ. 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 dλ for all Borel sets A. A f is the Radon-Nikodym derivative of µ w.r.t. λ, denoted f = dµ dλ. Our problem computable from µ? Is dµ dλ

5 Theorem On [0, 1], there is a computable measure µ λ (even µ 2λ) such that dµ dλ is not L1 (λ)-computable.

6 Theorem On [0, 1], there is a computable measure µ λ (even µ 2λ) such that dµ dλ is not L1 (λ)-computable. Proof. The measure will be defined as µ(a) = λ(a K) = λ(a K) λ(k) where K [0, 1]: is a recursive compact set, λ(k) > 0 is not computable (only upper semi-computable, or right-c.e.). I ɛ I 0 I 1 I 00 I 01 I 10 I 11 K

7 Theorem On [0, 1], there is a computable measure µ λ (even µ 2λ) such that dµ dλ is not L1 (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. dµ dλ = 1 λ(k) 1 K is not L 1 (λ)-computable. I ɛ I 0 I 1 I 00 I 01 I 10 I 11 K

8 (Non-)computability of RN Theorem There exists a computable measure µ λ such that dµ dλ is not L 1 (λ)-computable. Question How much is the Radon-Nikodym theorem non-computable?

9 (Non-)computability of RN Theorem There exists a computable measure µ λ such that dµ dλ 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.

10 (Non-)computability of RN Theorem There exists a computable measure µ λ such that dµ dλ 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.

11 Weihrauch degrees W is due to Weihrauch (1992).

12 Weihrauch degrees W is due to Weihrauch (1992). According to Klaus Weihrauch, W in W stands for Wadge.

13 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.

14 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.

15 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.

16 Consider two representations En and Cf of 2 N : En(p) = {n N : 100 n 1 is a subword of 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.

17 Properties of EC 2 0 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 ( ).

18 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 ( ).

19 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 ( ).

20 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 ( ).

21 EC W RN Let RN be the Radon-Nikodym operator, that maps µ λ to dµ dλ L1 (λ). Corollary EC W RN. Proof. Given an enumeration of E N: 1 construct K E such that λ(k E ) = n / E 2 n, 2 apply RN to compute λ(k E ), 3 compute E from λ(k E ). enumeration of E K E µ E EC characteristic function of E λ(k E ) dµ E dλ RN

22 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 dµ. It gives g L 2 (λ + µ) such that for all f L 2 (λ + µ), i.e. φ µ (f ) = f, g, f dµ = fg d(λ + µ). show that g 1 g has the required properties for being dµ dλ.

23 To compute the Radon-Nikodym derivative, dµ dλ = g µ RN φ µ FR 1 g L1 (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 dλ = 1. (a simple proof can be obtained using Martin-Löf randomness!)

24 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