## 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]
