Correspondence of Lebesgue-Stieltjes Measures and Distribution Functions

Last updated Nov 1, 2022

If $F: \mathbb{R} \to \mathbb{R}$ is a Distribution Function, then there is a unique Lebesgue-Stieltjes Measure $\mu_{F}$ on $\mathbb{R}$ such that $\mu_{F}((a,b]) = F(b) - F(a)$ for all $b > a \in \mathbb{R}$. If $G$ is another such Function, then $F - G$ is a constant.

Conversely, if $\mu$ is a Lebesgue-Stieltjes Measure and we define

$$F(x) = \begin{cases} -\mu((x, 0]) & x < 0 \\ 0 & x = 0 \\ \mu((0, x]) & x > 0 \\ \end{cases} $$ then $F$ is a Distribution Function and $\mu_{F}=\mu$.

TODO

We can define the following Categorys:

1. Let $\mu$ a be a Lebesgue-Stieltjes Measure on $\mathbb{R}$. Then construct the Category with

   1. Objects: elements of $\mathbb{R}$.
   2. Morphisms: Let $a, b \in \mathbb{R}$. The Morphism from $a$ to $b$ is
      1. If $a < b$, then the morphism is $\mu((a, b])$.
      2. If $a = b$, then the morphism is $\mu(\emptyset) = 0$.
      3. If $b < a$, then the morphism is $-\mu((b, a])$.

   Then Composition of Morphisms commutes because Measures are $\sigma$-additive. That is if we have $a, b, c \in \mathbb{R}$, then composing their morphisms by addition gets us the morphism from $a$ to $c$. Because $\mu$ is a Lebesgue-Stieltjes Measure, we have that each morphism is finite.

2. Let $F$ be a Distribution Function. Then construct the Category with

   1. Objects: elements of $\mathbb{R}$.
   2. Morphisms: Let $a, b \in \mathbb{R}$. The Morphism from $a$ to $b$ is $F(b) - F(a)$. This obviously composes as addition.

We can naturally transform from one to the other. The Non-Decreasing Function monotonicity of $F$ ensures that $\mu$ is non-negative on all sets. Caratheodory’s Theorem allows us to take the Category of $\mu$ to a unique Complete Measure (since $\mu$ will be Sigma-Finite Measure).

1. Observe that $G$ must be a Distribution Function since the Set of Distribution Functions is closed under addition of constants.

1. Folland - Real Analysis - pg 35

• Real Numbers
• Sigma Additive