Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law

1. (Uniqueness) Let $(X,\mathcal{S})$ be a measurable space, let $\mathcal{P}$ be a $\pi$-system of subsets of $X$ whose generated $\sigma$-algebra is $\mathcal{S}$, and let $\mu$ and $\nu$ be measures on $\mathcal{S}$ with $\mu(X)=\nu(X)<\infty$ and $\mu(A)=\nu(A)$ for every $A\in\mathcal{P}$. Then $\mu=\nu$.

2. (Rays generate the Borel $\sigma$-algebra) Each of the families $\{(u,\infty):u\in\mathbb{R}\}$ and $\{(-\infty,u]:u\in\mathbb{R}\}$ of subsets of the real numbers is a $\pi$-system whose generated $\sigma$-algebra is the Borel $\sigma$-algebra.

3. (Density of the exponential law) Let $\xi$ be a random variable on a probability space $(\Omega,\mathcal{F},P)$ with values in $[0,\infty)$ such that $P(\xi>u)=e^{-u}$ for every real $u\ge0$, where $e$ denotes the real exponential function. Then the distribution of $\xi$ is the measure with density $h$ with respect to Lebesgue measure on the Borel $\sigma$-algebra, where $h(u)=e^{-u}$ for $u>0$ and $h(u)=0$ for $u\le0$.