# An Integral Version of Hardy's Inequality

## TL;DR

This paper introduces a generalized form of Hardy's inequality applicable to measure spaces with a measurable function replacing the absolute value or distance, broadening the inequality's scope.

## Contribution

It presents an integral version of Hardy's inequality that extends classical results to more general measure spaces and functions.

## Key findings

- Generalizes Hardy's inequality to measure spaces with measurable functions
- Replaces absolute value or distance with a measurable function in the inequality
- Applicable to metric spaces with a distance function from a point

## Abstract

In this note we present a version of Hardy's inequality on a measure space $(X,\mu)$ endowed with a measurable function $N\colon X\to \mathbb R$ which replaces the absolute value on $\mathbb R$ or $\mathbb R^n$, and, more generally, the distance function from a given point when $X$ is a metric space.