Can the $L^1$-$L^\infty$ duality be restored for non-dominated families of probability measures?

TL;DR

This paper demonstrates how to restore the duality between $L^{ ext{infinity}}$ and its dual in non-dominated probability measure families by extending the probability space, enabling robust statistical modeling.

Contribution

It introduces a canonical extension of the probability space that restores duality for a broad class of non-dominated models, unifying existing frameworks.

Findings

Duality is restored on the extended model space.

The extension is minimal and canonical, preserving the original $\sigma$-algebra.

The framework applies to various models including Gaussian, Black-Scholes, and parametric models.

Abstract

The duality $L^{\infty}\simeq (L^{1})'$ frequently breaks down in the presence of model uncertainty, where a single reference measure $P$ is replaced by a non-dominated family of probability measures $\mathcal{P}$. The unavailability of classical measure-theoretic and functional-analytic tools in this regime poses a significant obstacle to developing robust probabilistic frameworks. We show that this duality can be restored for a broad class of robust statistical models by extending the underlying probability space. Specifically, on the extended model, the space $\mathbb{L}^{\infty}(\mathcal{P})$ of $\mathcal{P}$-quasi-surely bounded functions is isometrically isomorphic to the dual of the space of finite signed measures absolutely continuous with respect to at least one element of $\mathcal{P}$. The proposed extension is canonical: it is the smallest $\mathcal{P}$-complete extension of the original $\sigma$-algebra for which $\mathbb{L}^{\infty}(\mathcal{P})$ is the dual of any normed space. Our assumptions encompass several prominent non-dominated settings, including infinite product measures, Gaussian processes, the Black-Scholes model with uncertain constant volatility and drift, robust binomial models, and, more generally, infinite sequences from any parametric model with almost surely estimable parameters. Furthermore, we unify the existing frameworks of Cohen (2012) and Liebrich et al. (2022), demonstrating that our construction is equivalent to the capacity-based approach under mild assumptions satisfied by the aforementioned examples. Finally, we apply our theory to extend Kraft's (1955) characterization of strictly unbiased hypothesis tests to non-dominated cases.