Proof using Monotone Class Theorem

Question

As you know I have been grappling with this question since days ago, which I copy down here for convenience:

Let $X$ be set of $\mathbb R$, and let $\mathcal B$ be its Borel $\sigma$-algebra, and finally let $\mu_1$ and $\mu_2$ be the two measures on $(X,\mathcal B)$ such that $\mu_1((a,b))= \mu_2((a,b)) < \infty$ whenever $−\infty < a < b < \infty$. Show that $\mu_1(A) = \mu_2(A)$ whenever $A \in \mathcal B$.​

As you know again that this question comes from early chapters of Richard F. Bass's introductory book, therefore any solution should involve no advanced theorems such as Dynkin's. The consensus I got so far is to use the Monotone Class Theorem:

Suppose $\mathscr A_0$ is an algebra, $\mathscr A$ is the smallest $\sigma$-algebra containing $\mathscr A_0$, and $\mathscr M$ is the smallest monotone class containing $\mathscr A_0$, then $\mathscr A = \mathscr M$.

Here are my three questions:

(1) My proof idea (naive, perhaps) is to show that, $$\begin{align} \text{if } A \in \mathcal B &\text{ then } A \in \mathscr A \\ \text{if } (a, b) \in \mathbb R^1 &\text{ then } (a, b) \in \mathscr M. \\ \end{align}$$

Since by the Monotone Theorem $\mathscr A = \mathscr M$, therefore I can arrive at the result. Is this idea so flawed that it is dead on arrival? (Who knows, since breakthroughs sometimes happen when someone asks a really dumb question. :-) )

(2) If my idea is valid, how do you prove that if $A \in \mathcal B \rightarrow A \in \mathscr A$? I mean the mechanic of going from $A \in \mathcal B$ to $ A \in \mathscr A$?

(3) And finally, what is the mechanics of proving that if $(a, b) \in \mathbb R^1 \rightarrow (a, b) \in \mathscr M$?

Thank you for your time.

---

POST SCRIPT! POST SCRIPT!: I finally came up with solution without any advanced theorems, even the Monotone Theorem, adapted from a solution by @JoshKeneda, who had used Dynkin's Theorem. My solution is posted here at the answer page down below. I have submitted this work to my professor, he ok'd it except for (5) because it is true only when the $A_i$'s are pairwise disjoint. Feel free to drop me a message if you have ideas to improve (5). Thanks to all and especially to @JoshKeneda.

Answer 1

You don't even need the monotone class theorem (that is at least as advanced as Dynkin-stuff).

First extend the property of $\mu_1$ and $\mu_2$ coinciding, to half-open intervals $(a,b]$ as follows:

Pick $c>b$. Then use the fact that $\mu_1(a,c) = \mu_2(a,c)$ and $\mu_1(b,c) = \mu_2(b,c)$, together with additivity of measures to get the result. Then do this with the other kind of half-open intervals

Using continuity of measures, you can also prove the same holds for open and closed rays, since a ray is just an increasing union of intervals.

Now make an algebra $\mathscr A_0$ (closed under unions and complements) out of your choice of basic intervals, which generates the Borel $\sigma$-algebra. In other words, we are applying the standard argument for showing that all Borel sets satisfy some property:

1) Show that the collection $\mathscr{B}_+$ of Borel sets on which $\mu_1=\mu_2$ is a $\sigma$-algebra.

2) You showed that $\mathscr A_0$ belongs to $\mathscr{B}_+$, hence $$\mathscr{B} = \sigma(\mathscr{A}_0) \subset \mathscr{B}_+$$