Doubt in understanding extension theorem

Question

I was reading G Folland Measure theory book. I incured proof of extension theorem. I think I had understand the proof. But I do not understand one thing.

I understand a part

But For yellow high lighted step Auther take intersection with A

I thought this may be due to he had assumed that A is measurable and by Cartheodary Criterion

$\mu_*(B_j)=\mu_*(B_j\cap A)+\mu_*(B_j\cap A^c)$

Is this correct? Or Auther have another argument?

Please help me to understand this proof.

Answer 1

Since $A\in\mathcal{A}$ and $\{B_j\}\subset \mathcal{A}$, $B_j\cap A\in\mathcal{A}$ and $B_j\cap A^c\in\mathcal{A}$. $\mu_0$ is finitely additive on $\mathcal{A}$ and, thus, $$ \mu_0(B_j)=\mu_0(B_j\cap A)+\mu_0(B_j\cap A^c). $$ Consequently, $$ \mu^{*}(E)+\epsilon\ge \sum_{j\ge 1}\mu_0(B_j)=\sum_{j\ge 1}\mu_0(B_j\cap A)+\mu_0(B\cap A^c). $$