# The distribution function of a probability measure on a linearly ordered topological space

## TL;DR

This paper develops a theory of cumulative distribution functions on linearly ordered topological spaces, extending classical concepts to more general settings and enabling sampling and integration with respect to such measures.

## Contribution

It introduces a generalized distribution function and its pseudo-inverse in ordered topological spaces, providing tools for sampling and integration in these contexts.

## Key findings

- Defined a distribution function for probability measures on ordered topological spaces
- Established properties of the pseudo-inverse of the distribution function
- Enabled sampling and integral calculation for these measures

## Abstract

In this paper we describe a theory of a cumulative distribution function on a space with an order from a probability measure defined in this space. This distribution function plays a similar role to that played in the classical case. Moreover, we define its pseudo-inverse and study its properties. Those properties will allow us to generate samples of a distribution and give us the chance to calculate integrals with respect to the related probability measure.