Extension of a Finite Premeasure from an Algebra of Sets to the Generated Sigma-Algebra

In the setting of The Real Numbers: Standing Notation and Background, let $\Omega$ be a set and let $\mathcal{A}$ be a family of subsets of $\Omega$ such that $\Omega\in\mathcal{A}$, $\Omega\setminus A\in\mathcal{A}$ for every $A\in\mathcal{A}$, and $A\cup B\in\mathcal{A}$ for all $A,B\in\mathcal{A}$; in particular $\varnothing=\Omega\setminus\Omega\in\mathcal{A}$. Let $\sigma(\mathcal{A})$ be the $\sigma$-algebra generated by $\mathcal{A}$, a $\sigma$-algebra on $\Omega$ by claim 2 of Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra, so that $(\Omega,\sigma(\mathcal{A}))$ is a measurable space. Let $m:\mathcal{A}\to\mathbb{R}$ be a function with $m(A)\ge0$ for every $A\in\mathcal{A}$ and $m(\varnothing)=0$ which is countably additive on $\mathcal{A}$: for every sequence $(A_{i})_{i\in\mathbb{N}}$ of pairwise disjoint members of $\mathcal{A}$ whose union $A$ belongs to $\mathcal{A}$, the series $\sum_{i=1}^{\infty}m(A_{i})$ converges and its sum is $m(A)$.

1. (Extension)¶ There is a measure $\bar{m}$ on the measurable space $(\Omega,\sigma(\mathcal{A}))$ with $\bar{m}(A)=m(A)$ for every $A\in\mathcal{A}$; in particular $\bar{m}(\Omega)=m(\Omega)$ is a real number, so that $\bar{m}$ is finite.

2. (Uniqueness)¶ If $\bar{m}_{1}$ and $\bar{m}_{2}$ are measures on $(\Omega,\sigma(\mathcal{A}))$ with $\bar{m}_{1}(A)=\bar{m}_{2}(A)=m(A)$ for every $A\in\mathcal{A}$, then $\bar{m}_{1}=\bar{m}_{2}$.