[Paper Review] Computability of the Radon-Nikodym derivative

Mathieu Hoyrup, Cristóbal Rojas+1 more|arXiv (Cornell University)|Dec 13, 2011

Computability, Logic, AI Algorithms25 references4 citations

TL;DR

This paper investigates the computability of the Radon-Nikodym derivative in the framework of Type Two Effectivity (TTE). It establishes that the Radon-Nikodym operator is W-reducible to the non-computable operator EC (mapping enumerations to characteristic functions), and constructs a computable measurable space where EC is W-reducible to the Radon-Nikodym operator, showing they are computationally equivalent in this setting.

ABSTRACT

We study the computational content of the Radon-Nokodym theorem from measure theory in the framework of the representation approach to computable analysis. We define computable measurable spaces and canonical representations of the measures and the integrable functions on such spaces. For functions f,g on represented sets, f is W-reducible to g if f can be computed by applying the function g at most once. Let RN be the Radon-Nikodym operator on the space under consideration and let EC be the non-computable operator mapping every enumeration of a set of natural numbers to its characteristic function. We prove that for every computable measurable space, RN is W-reducible to EC, and we construct a computable measurable space for which EC is W-reducible to RN.

Motivation & Objective

• To analyze the computational content of the Radon-Nikodym derivative in computable measure theory.
• To define computable measurable spaces with canonical representations for measures and integrable functions.
• To characterize the degree of non-computability of the Radon-Nikodym operator using W-reducibility.
• To establish a computational equivalence between the Radon-Nikodym operator and the EC operator on certain computable measurable spaces.

Proposed method

• Using the representation approach in computable analysis (TTE), the paper defines computable measurable spaces with representations for σ-finite and finite measures and integrable functions.
• The Radon-Nikodym operator RN is analyzed via W-reducibility, where f ≤_W g means f can be computed with a single application of g.
• The proof of RN ≤_W EC relies on Levy’s zero-one law and the classical Radon-Nikodym theorem applied in a computable setting.
• A specific computable measurable space is constructed where EC ≤_sW RN, demonstrating a tight computational equivalence.
• The construction uses a sequence of step functions converging to the Radon-Nikodym derivative, with EC used to extract a fast-converging subsequence.
• Alternative proofs via the Fréchet-Riesz representation theorem are considered, with effectivization requiring a single application of EC to achieve fast convergence.

Experimental results

Research questions

• RQ1 Can the Radon-Nikodym derivative be computed from the measures μ and λ in a computable setting?
• RQ2 What is the degree of non-computability of the Radon-Nikodym operator in terms of W-reducibility?
• RQ3 Is there a computable measurable space for which the Radon-Nikodym operator computes the EC operator?
• RQ4 Can the classical Radon-Nikodym theorem be effectivized using only computable operations and a single application of EC?
• RQ5 Under what conditions can the Radon-Nikodym derivative be computed without using EC?

Key findings

• For every computable measurable space, the Radon-Nikodym operator RN is W-reducible to the EC operator, i.e., RN ≤_W EC.
• There exists a simple computable measurable space for which EC is W-reducible to RN, i.e., EC ≤_sW RN.
• In this space, the Radon-Nikodym operator and the EC operator are computationally equivalent: RN ≡_W EC.
• The proof of RN ≤_W EC uses Levy’s zero-one law and the classical Radon-Nikodym theorem in a computable framework.
• The construction of the counterexample space relies on a sequence of step functions whose convergence is controlled via EC to extract a fast-converging subsequence.
• Alternative effectivizations via the Fréchet-Riesz representation theorem require a single application of EC to achieve fast convergence, even when the norm is computable from below.