Consider the field $\beta_0$ which contains all the finite disjoint unions of intervals within $(0,1]$. For example, the generic element can be expressed as $A_k=\cup_{i=1}^nI_i$ where the $I_i$'s are disjoint intervals. My textbook offers a proof that Lebesgue measure $\lambda$ is countably additive over the field $\beta_0$, and I believe the proof uses an unjustified assumption.
In the proof, it is written: "Suppose that $A=\cup_{i=1}^{\infty}A_i$, where the $A_i$'s are disjoint, and $A$ and the $A_i$'s are $\beta_0$-sets."
My question: how do we know that $A$ is a $\beta_0$-set? Although $\beta_0$ is a field, it is not a $\sigma$-field.
Edit: Problem solved in the comments.