Theorem (Radon-Nikodym Derivative is the Density)

Let $I=[a,b]$. Let $F:I\to \mathbb{R}$ be Continuous and non-decreasing. Then the following are equivalent:

1. $F$ is absolutely continuous on $[a,b]$
2. $F$ maps sets of Lebesgue Measure zero to sets of Lebesgue measure zero
3. $F$ is differentiable $m$-a.e. with $F'\in L^{1}(m)$ and $$\forall x \in [a,b]:F(x)-F(a)=\int\limits _{[a,x]}F' \, dm$$

Theorem (Almost every point is a Lebesgue point)

Let $f\in L^{1}(\mathbb{R})$. Then almost every $x \in\mathbb{R}$ is a for $f$.

Theorem

Suppose that for every $x \in\mathbb{R}$ we have a sequence $(E_{j}(x))_{j\ge 1}$ that to $x$. Let $f\in L^{1}(\mathbb{R})$. Then for every $x$ of $f$ we have that $$f(x)=\lim_{ j \to \infty } \frac{1}{m(E_{j}(x))}\int\limits _{E_{j}(x)}f \, dm$$

Theorem (Density is the Radon-Nikodym Derivative)

Suppose that $f\in L^{1}(\mathbb{R})$. Define, $\forall x \in\mathbb{R}$: $$F(x)=\int\limits _{[-\infty,x]}f \, dm$$ Then for every $x$ of $f$ we have $$F'(x)=f(x)$$