## Introduction

In measure theory, we often want to compare two measures $\nu$ and $\mu$ to see how “thick” or “thin” one is relative to the other at each point. Classical theorems like Radon–Nikodym guarantee a global derivative $d\nu/d\mu$ when $\nu$ is absolutely continuous with respect to $\mu$, but they say nothing about recovering that derivative **pointwise** or handling singular parts. To fill this gap, we introduce the **upper** and **lower** pointwise densities of $\nu$ relative to $\mu$. These densities:

- Capture the local “growth rate” $\nu(B(x,r))/\mu(B(x,r))$ as $r\to 0$. - Allow us to characterize absolute [continuity](/page/Continuity) and singularity in a fine, pointwise way. - Lead directly to the Lebesgue–Besicovitch differentiation theorem, which reconstructs $d\nu/d\mu$ almost everywhere and splits off the singular part.

With these definitions in hand, one can rigorously “differentiate” measures and recover classical decomposition results in a metric-measure setting.

## 1. Pointwise Densities

[definition] For each point $x\in\mathbb{R}^n$, define the **upper** and **lower** derivatives of a Radon measure $\nu$ with respect to another Radon measure $\mu$ by \begin{align*} \overline D_{\mu}\nu(x) &:=\limsup_{r\to 0} \frac{\nu\bigl(B(x,r)\bigr)}{\mu\bigl(B(x,r)\bigr)} \quad\text{if }\mu\bigl(B(x,r)\bigr)>0\text{ for all }r>0, \\ &:=+\infty\quad\text{if }\mu\bigl(B(x,r)\bigr)=0\text{ for some }r>0, \end{align*} and \begin{align*} \underline D_{\mu}\nu(x) &:=\liminf_{r\to 0} \frac{\nu\bigl(B(x,r)\bigr)}{\mu\bigl(B(x,r)\bigr)} \quad\text{if }\mu\bigl(B(x,r)\bigr)>0\text{ for all }r>0, \\ &:=+\infty\quad\text{if }\mu\bigl(B(x,r)\bigr)=0\text{ for some }r>0. \end{align*} [/definition]

[definition] If $\overline D_{\mu}\nu(x)=\underline D_{\mu}\nu(x)<+\infty$, we say $\nu$ is **differentiable** with respect to $\mu$ at $x$ and write \begin{align*} D_{\mu}\nu(x):=\overline D_{\mu}\nu(x)=\underline D_{\mu}\nu(x). \end{align*} This value is also called the **density** of $\nu$ with respect to $\mu$ at $x$. [/definition]

## 2. Relation to Radon–Nikodym

[theorem: Radon–Nikodym Theorem] If $\nu\ll\mu$, then there exists a function $f\in L^1(\mu)$ such that for every measurable set $E$, \begin{align*} \nu(E)=\int_E f\,d\mu. \end{align*} The function $f$ is unique up to $\mu$–null [sets](/page/Set) and is denoted $d\nu/d\mu$. [/theorem]

## 3. [Differentiation](/page/Derivative) of Measures

[theorem] **Lebesgue–Besicovitch Differentiation Theorem.** Let $\mu$ and $\nu$ be Radon measures on $\mathbb{R}^n$. Then for $\mu$–almost every $x$: \begin{align*} D_{\mu}\nu(x)=\frac{d\nu}{d\mu}(x) \quad\text{and}\quad \overline D_{\mu}\nu(x)<+\infty. \end{align*} Moreover, one has the decomposition \begin{align*} \nu(E) =\int_E D_{\mu}\nu(x)\,d\mu(x)+\nu_s(E), \end{align*} where $\nu_s\perp\mu$ is the singular part of $\nu$. [/theorem]

## 4. Key Lemmas

[lemma] Fix $0<\alpha<\infty$. Then for any set $A\subset\mathbb{R}^n$ (not necessarily measurable): 1. If $\{x\in A:\underline D_{\mu}\nu(x)\le\alpha\}=A$, then $\nu(A)\le\alpha\,\mu(A)$. 2. If $\{x\in A:\overline D_{\mu}\nu(x)\ge\alpha\}=A$, then $\nu(A)\ge\alpha\,\mu(A)$. [/lemma]

## 5. Remarks and Applications

[remark] The [functions](/page/Function) $\overline D_{\mu}\nu$ and $\underline D_{\mu}\nu$ provide a **pointwise** characterization of absolute continuity and singularity: - If $\overline D_{\mu}\nu(x)<\infty$ for $\mu$–a.e.\ $x$, then $\nu\ll\mu$. - If $\underline D_{\mu}\nu(x)=+\infty$ on a set of full $\nu$–measure, then $\nu\perp\mu$. [/remark]

[remark] In practice, these densities recover the intuitive notion of “local growth rate” of $\nu$ relative to $\mu$, and they play a crucial role in geometric measure theory, probability theory, and analysis of PDEs via variational methods. [/remark]