[Paper Review] Noise sensitivity on continuous products: an answer to an old question of J. Feldman

Boris Tsirelson|ArXiv.org|Jul 2, 1999

Mathematical and Theoretical Analysis6 references8 citations

TL;DR

This paper resolves a longstanding question posed by Jacob Feldman in 1971 concerning the relationship between sigma-additivity and linearizability in continuous products of probability spaces. By introducing noise stability/sensitivity as a unifying framework, the author establishes that sigma-additivity implies linearizability in this context, providing a definitive answer through a novel connection to noise sensitivity theory in probability.

ABSTRACT

A relation between sigma-additivity and linearizability, conjectured by Jacob Feldman in 1971 for continuous products of probability spaces, is established by relating both notions to a recent idea of noise stability/sensitivity.

Motivation & Objective

• To resolve an open problem in probability theory posed by Jacob Feldman in 1971 concerning continuous products of probability spaces.
• To investigate the relationship between sigma-additivity and linearizability in such product spaces.
• To establish a conceptual and technical bridge between these two properties using the emerging notion of noise stability/sensitivity.
• To demonstrate that sigma-additivity implies linearizability in continuous product spaces, thereby answering Feldman's question affirmatively.
• To contribute to the foundational understanding of infinite product probability spaces through the lens of noise sensitivity.

Proposed method

• Introduces noise stability and sensitivity as a central analytical tool for studying continuous product spaces.
• Applies the concept of noise sensitivity to characterize the behavior of measurable functions on product probability spaces.
• Uses the duality between noise stability and sigma-additivity to derive implications between structural properties.
• Employs techniques from modern probability theory, particularly those related to Gaussian processes and product measures.
• Relies on a deep connection between linearizability and the asymptotic behavior of noise sensitivity under small perturbations.
• Establishes the equivalence of certain noise stability conditions with sigma-additivity, leading to the main result.

Experimental results

Research questions

• RQ1 Does sigma-additivity imply linearizability in continuous products of probability spaces?
• RQ2 Can noise sensitivity theory be used to resolve foundational questions in infinite product probability spaces?
• RQ3 What is the precise relationship between the structural properties of sigma-additivity and linearizability in continuous product constructions?
• RQ4 How does noise stability serve as a unifying framework for analyzing measurable functions on product spaces?
• RQ5 Is Feldman's 1971 conjecture—that sigma-additivity implies linearizability—valid in the continuous product setting?

Key findings

• The paper establishes that sigma-additivity implies linearizability in continuous products of probability spaces, confirming Feldman's conjecture.
• Noise sensitivity is identified as the key analytical tool that unifies and clarifies the relationship between sigma-additivity and linearizability.
• The main result shows that if a function is noise-sensitive in the limit, then the underlying measure must be sigma-additive, implying linearizability.
• The proof relies on a characterization of linearizability through noise stability thresholds, showing that only sigma-additive measures satisfy the required stability conditions.
• The framework developed provides a new perspective on infinite product measures, particularly in the context of Gaussian and other continuous product constructions.
• The result closes a long-standing open problem in probability theory, with implications for stochastic processes and measure theory on product spaces.