A co-analytic maximal set of orthogonal measures

TL;DR

This paper proves that under the assumption V=L, there exists a highly definable maximal orthogonal set of measures on Cantor space, contrasting with the non-existence of such sets among analytic measures.

Contribution

It establishes the existence of a a1^1_1 maximal orthogonal set of measures under V=L, providing a new example in descriptive set theory.

Findings

Existence of a a1^1_1 maximal orthogonal set under V=L

Contrasts with the non-existence of analytic maximal orthogonal sets

Highlights the role of set-theoretic assumptions in measure theory

Abstract

We prove that if $V=L$ then there is a $\Pi^1_1$ maximal orthogonal (i.e. mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known Theorem of Preiss and Rataj that no analytic set of measures can be maximal orthogonal.