Proof of Properties of Distribution Functions

Throughout, $F(x)=\mu((-\infty,x])$ for $x\in\mathbb{R}$ (Distribution Functions (i)), the rays $(-\infty,x]$ are Borel sets (Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii)), and $\mu(\emptyset)=0$, $\mu(\mathbb{R})=1$ (Measure, Measure Space, and Probability Measure (i), Measure, Measure Space, and Probability Measure (v)); the field and order laws of $\mathbb{R}$ are used freely (The Structure of ℝ as a Field and an Ordered Ring (i), The Structure of ℝ as a Field and an Ordered Ring (ii), Rules of Arithmetic and Order in an Ordered Field (iv), Rules of Arithmetic and Order in an Ordered Field (v), Rules of Arithmetic and Order in an Ordered Field (viii), The Quotient 1/x, Order Rules for Sums, and Nesting of Closed Intervals in ℝ (i)).

Monotonicity. For $x\le y$ we have $(-\infty,x]\subseteq(-\infty,y]\subseteq\mathbb{R}$, so $F(x)\le F(y)\le\mu(\mathbb{R})=1$ by Basic Properties of a Measure (iii); and $F(x)\ge0$ since $\mu$ takes values in $[0,\infty]$.

Right-continuity. Fix $x$ and $\varepsilon$, and let $A_{k}=(-\infty,x+1/k]$ for $k\in\mathbb{N}$. The sets $A_{k}$ decrease, $\mu(A_{1})\le1$, and

$$\bigcap_{k\in\mathbb{N}}A_{k}=(-\infty,x],$$

since both inclusions hold: every $A_{k}$ contains $(-\infty,x]$ because $x<x+1/k$; and if $t>x$, there is $k\in\mathbb{N}$ with $1/k<t-x$ (The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (iii)), so $t\notin A_{k}$. By continuity from above (Basic Properties of a Measure (ix)), $F(x)=\inf_{k}F(x+1/k)$. As $F(x)<F(x)+\varepsilon$, there is $k$ with $F(x+1/k)<F(x)+\varepsilon$ (Order and Suprema in the Extended Real Line (x)). Put $\delta=1/k$. For $x\le y<x+\delta$, (i) gives

$$F(x)\le F(y)\le F(x+1/k)<F(x)+\varepsilon .$$

Limit at $-\infty$. The sets $(-\infty,-k]$, $k\in\mathbb{N}$, decrease, $\mu((-\infty,-1])\le1$, and their intersection is empty, since every $t\in\mathbb{R}$ satisfies $-t<k$ for some $k\in\mathbb{N}$ (The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (ii)). By continuity from above (Basic Properties of a Measure (ix)), $\inf_{k}F(-k)=\mu(\emptyset)=0$, so there is $k\in\mathbb{N}$ with $F(-k)<\varepsilon$ (Order and Suprema in the Extended Real Line (x)). Put $M=k\ge1$. By (i), $F(x)\le F(-M)<\varepsilon$ for $x\le-M$.

Limit at $\infty$. The sets $(-\infty,k]$, $k\in\mathbb{N}$, increase and their union is $\mathbb{R}$, since every $t\in\mathbb{R}$ satisfies $t<k$ for some $k\in\mathbb{N}$ (The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (ii)). By continuity from below (Basic Properties of a Measure (viii)), $\sup_{k}F(k)=\mu(\mathbb{R})=1$, so there is $k\in\mathbb{N}$ with $F(k)>1-\varepsilon$ (Order and Suprema in the Extended Real Line (x)). Put $M=k\ge1$. By (i), $F(x)\ge F(M)>1-\varepsilon$ for $x\ge M$.

The distribution function determines the measure. Let $\mathcal{E}=\{(-\infty,x]:x\in\mathbb{R}\}$. Since

$$(-\infty,x]\cap(-\infty,y]=\bigl(-\infty,\min\{x,y\}\bigr],$$

$\mathcal{E}$ is a $\pi$-system on $\mathbb{R}$ (π-System (i)), and $\sigma(\mathcal{E})=\mathcal{B}(\mathbb{R})$ by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (iii). The measures $\mu$ and $\nu$ agree on $\mathcal{E}$, because $\nu((-\infty,x])=F_{\nu}(x)=F_{\mu}(x)=\mu((-\infty,x])$, and $\mu(\mathbb{R})=\nu(\mathbb{R})=1<\infty$. Hence $\mu=\nu$ by Uniqueness of Measures Agreeing on a Generating π-System (i).

Identically distributed random variables. As $X$ and $Y$ are random elements of $\mathbb{R}$ (Probability Spaces, Events and Random Elements (iv)), their laws $P_{X}$ and $P'_{Y}$ are probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, with $F_{X}=F_{P_{X}}$ and $F_{Y}=F_{P'_{Y}}$ (The Law of a Random Element (i), Distribution Functions (ii)). By The Law of a Random Element (ii), $X$ and $Y$ are identically distributed exactly when $P_{X}=P'_{Y}$. If $P_{X}=P'_{Y}$, then $F_{X}=F_{Y}$; conversely, if $F_{X}=F_{Y}$, then $P_{X}=P'_{Y}$ by (v).