Product measure without integration

Question

Is it possible to prove the existence of the product measure without the concept of integration?

My thoughts are focused on trying to prove it indirectly using the Carathéodory's extension theorem. The path I'm trying to follow is described bellow:

Let $(\Omega_1,\mathcal{F}_1,\mu_1)$ and $(\Omega_2,\mathcal{F}_2,\mu_2)$ be two measure spaces. If $A\times B\subset\Omega_1\times\Omega_2$ is a rectangle, that is, $A\in\mathcal{F}_1$ and $B\in\mathcal{F}_2$ such that $A\times B=\bigoplus_{i=1}^{\infty}A_i\times B_i$ where $A_i\times B_i$ is a rectangle for every $i\in\mathbb{N}$ and $\bigoplus$ stands for disjoint union.

1. Can we always obtain a (potentially non countable!!) regular subdivision of $A\times B$. That is, can we always write $A\times B=\bigoplus_{j\in J}\bigoplus_{k\in K}A_j'\times B_k'$, being $A_i'\times B_j'$ rectangles such that $A=\bigoplus_{j\in J}A_j'$, $B=\bigoplus_{k\in K}B_k'$, $A_i=\bigoplus_{j\in J(A_i)} A_j'$ and $B_j=\bigoplus_{k\in K(B_j)} B_k'$.

We just have to consider the family $\mathcal{C}_A:=\{\bigcap A_i\not=\emptyset\}$ (analogously for $B$), then define for every $C\in\mathcal{C}_A$ (analogously for $\mathcal{C}_B$) $C':=\bigcup_{C\not\subset A_i}A_i$, and finally consider $\mathcal{D}_A:=\{C\setminus C':C\in\mathcal{C}_{A}\}$ (analogously for $B$). Then it is easy to prove that the products $A_j\times B_k$ where $A_j\in\mathcal{D}_A$ and $B_k\in\mathcal{D}_{B}$ make up a regular subsivision of $A\times B$.

2. In order to apply Carathéodory's extension theorem we need to prove that if $A\times B=\bigoplus_{i=1}^{\infty}A_i\times B_i$ then $\mu(A\times B)\leq \sum_{i=1}^{\infty}\mu(A_i\times B_i)$ where $\mu$ is the hypothetical product measure. For the finite case we can obtain the equality easily, but for the infinite case I really don't know what to do when the regular subdivision is non countable.

I suspect that my last bold remark is the reason why I've never came across with the required proof, but maybe there is some trick that I'm missing...

Any ideas?