Uniqueness of Measures Agreeing on a Generating π-System

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $(X,\mathcal{F})$ be a measurable space, let $\mathcal{E}$ be a $\pi$-system on $X$ such that the $\sigma$-algebra generated by $\mathcal{E}$ is $\sigma(\mathcal{E})=\mathcal{F}$, so that $\mathcal{E}\subseteq\mathcal{F}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), and let $\mu$ and $\nu$ be measures on $(X,\mathcal{F})$ with $\mu(E)=\nu(E)$ for every $E\in\mathcal{E}$. Here $\infty$ is the point of The Extended Real Line, Its Order, and [0,∞] (i), and $x<y$, for $x,y$ in $[0,\infty]$, is as in The Extended Real Line, Its Order, and [0,∞] (iii).

(i) If $\mu(X)=\nu(X)$ and $\mu(X)<\infty$, then $\mu(A)=\nu(A)$ for every $A\in\mathcal{F}$, that is, $\mu=\nu$.

(ii) If there is a sequence $(E_{m})$ in $\mathcal{E}$ whose union $\bigcup_{m\in\mathbb{N}}E_{m}$ is $X$ and with $\mu(E_{m})<\infty$ for every $m\in\mathbb{N}$, then $\mu(A)=\nu(A)$ for every $A\in\mathcal{F}$, that is, $\mu=\nu$.