外测度的有限次可加性：我的证明是否正确？求验证与优化建议

Hey everyone, I’ve been working through deriving the finite subadditivity of the outer measure on $\mathbb{R}^d$, and I’m hoping to get some community feedback here. Specifically, I’d love:

• A check on whether my proof reasoning is valid
• Suggestions to make the proof more rigorous or easier to follow
• Help polishing one specific step that I’m stuck on (I can share more details about the step if needed!)

Proof Goal #

Show that for any subsets $A_1, A_2, ..., A_n \subset \mathbb{R^d}$, the following holds:
$$m^(A_1 \cup A_2 \cup \cdots \cup A_n) \le m^*(A_1)+m(A_2) + \cdots + m^*(A_n)$$

Given Lemma (No Verification Needed) #

• Let $A, B \subset \mathbb{R}$, where $\inf A$ and $\inf B$ exist. Define $A+B = {a+b \mid a \in A, b \in B}$. Then $\cdots$ (rest of the lemma statement was omitted in the original query)

内容的提问来源于stack exchange，提问作者Ovi