Conditional expectation operators on $C(X)$

TL;DR

This paper constructs conditional expectation operators in the space of continuous functions on a compact space, expanding their applicability beyond traditional Lp spaces and contributing to the theory of integrals in C(X).

Contribution

It provides a positive construction of conditional expectation operators in C(X), addressing a question from the 2021 COSAEF conference and advancing the understanding of integrals in this setting.

Findings

Conditional expectation operators can be constructed in C(X).

The work extends the theory beyond Lp spaces.

It contributes to the discussion on integrals in C(X).

Abstract

At the COSAEF conference in 2021, several participants asked the question whether a conditional expectation operator in the sense of Kuo, Labaushagne and Watson could be constructed in vector lattices other than $\mathcal{L}_p$ spaces and in particular in $C(X)$. This work answers positively to this question and participates in an old discussion on integrals in $C(X)$ space.