A more general version of Folland, "Real Analysis", Theorem 1.14 [duplicate]

Let $ X $ be a set. Let $ \mathcal E $ be a semi-ring of ("elementary") subsets of $ X $, i.e. assume that

1. $ \emptyset\in \mathcal E $;
2. $ \mathcal E $ is closed for finite intersections;
3. the set difference $ F\setminus E $ between two members of $ \mathcal E $ can always be written as a finite union of disjoint members of $ \mathcal E $.

Let $ \rho\colon \mathcal E\to \left[0,+\infty\right] $ be a set function satisfying the following properties:

1. $ \rho(\emptyset) = 0 $
2. $ \rho $ is additive, i.e., $ \rho\Bigl(\bigcup_{j\in \mathbb N}E_j\Bigr) = \sum_{j\in \mathbb N}\rho(E_j) $ whenever $ \{E_j\} $ is a countable collection of disjoint members of $ \mathcal E $;
3. $ \rho $ is $ \sigma $-subadditive, i.e., $ \rho(E)\leqq \sum_{j\in \mathbb N}\rho(E_j) $ whenever $ E\in \mathcal E $ and $ \{E_j\} $ is a countable collection of (not necessarily disjoint) members of $ \mathcal E $ such that $ E\subset \bigcup_{j\in \mathbb N}E_j $.

Assume moreover that the entire space $ X $ can be covered by a countable family $ \{E_j\} $ of members of $ \mathcal E $ such that $ \rho(E_j) < +\infty $.

I'm trying to show that every measure on the $ \sigma $-algebra $ \langle \mathcal E\rangle_\sigma $ generated by $ \mathcal E $ that extends $ \rho $ is uniquely determined by the values it assumes on the members of $ \mathcal E $.

Folland's Real Analysis provide a cute proof of this fact but assumes stronger hypotheses[*], and it's argument can't directly be carried over to the present setting. I was wondering if it would be possible to find an elementary proof this fact along the lines of the one that appears in Folland's book without appealing to big theorems like $ \pi $-$ \lambda $'s or whatever.

[*] I.e., he assumes that $ \mathcal E $ is an algebra of subsets of $ X $ and $ \rho $ is a premeasure on $ X $. It can be shown that under these requirements $ \rho $ satisfy properties 1., 2. and 3.