The Giry Monad is not Strong for the Canonical Symmetric Monoidal Closed Structure on Meas

TL;DR

This paper demonstrates that the Giry monad does not possess the strength property within the standard symmetric monoidal closed structure on the category of measurable spaces, impacting its theoretical applications.

Contribution

It establishes a fundamental limitation of the Giry monad's compatibility with the canonical monoidal structure in measure theory.

Findings

Giry monad is not strong in the canonical structure

Implications for measure-theoretic categorical models

Clarifies limitations in monad applications to measurable spaces

Abstract

We show that the Giry monad is not strong with respect to the canonical symmetric monoidal closed structure on the category Meas of all measurable spaces and measurable functions.