# A metric version of Schlichting's Theorem

## TL;DR

This paper extends Schlichting's Theorem to metric structures, showing that for a type-definable family of commensurable objects, an invariant object exists that is commensurable with the family.

## Contribution

It introduces a metric version of Schlichting's Theorem applicable to type-definable families in metric structures, broadening its scope.

## Key findings

- Existence of invariant objects commensurable with type-definable families
- Applicability to classical first-order structures and hyper-definable objects
- Generalization of Schlichting's Theorem to metric settings

## Abstract

If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.