# The cut metric for probability distributions

## TL;DR

This paper explores a variant of the cut metric for probability distributions, introducing a pinning operation and demonstrating its properties and applications, including a regularity lemma analogue and continuity of operations.

## Contribution

It introduces a natural pinning operation on limit objects and establishes a canonical cut metric approximation for probability distributions, extending graph limit concepts.

## Key findings

- Pinning operation yields a canonical cut metric approximation.
- Cut metric continuity shown for product measures.
- Establishes basic properties of the cut metric for probability distributions.

## Abstract

Guided by the theory of graph limits, we investigate a variant of the cut metric for limit objects of sequences of discrete probability distributions. Apart from establishing basic results, we introduce a natural operation called {\em pinning} on the space of limit objects and show how this operation yields a canonical cut metric approximation to a given probability distribution akin to the weak regularity lemma for graphons. We also establish the cut metric continuity of basic operations such as taking product measures.