Measurable Functions and Topolgical Algebra

TL;DR

This paper demonstrates that measurable functions from a measurable space to a topological model of a Lawvere theory form a set-theoretic model of that theory, with implications for rings of real and complex measurable functions.

Contribution

It establishes a connection between measurable functions and models of Lawvere theories, providing new proofs for algebraic structures of measurable functions.

Findings

Meas(X,Y) forms a set-theoretic model of the Lawvere theory .

The set of real-valued measurable functions on X is a ring.

The set of complex-valued measurable functions on X is a ring.

Abstract

In this paper we show that if $(X,\mathcal{A})$ is a measurable space and if $Y$ is a topological model of a Lawvere theory $\mathcal{T}$ equipped with $\mathcal{B}$ the Borel $\sigma$-algebra on $Y$, then the set of $\mathcal{B}$-measurable functions from $X$ to $Y$, $\operatorname{Meas}(X,Y)$, is a set-theoretic model of $\mathcal{T}$. As a corollary we give short proofs of the facts that the set of real-valued measurable functions on a measurable space $X$ is a ring and the set of complex-valued measurable functions from $X$ to $\mathbb{C}$ is a ring.