Showing that $\mu(\phi^{-1}(V)) = \lambda (V).$

In a process of showing that the operator $T$ we gave here Showing that a set has measure zero? is ergodic, I want to prove that $\mu(\phi^{-1}(V)) = \lambda (V).$ where I have the following givens to prove it:

Suppose $V$ is a measurable subset of $[0,1).$ Let $\lambda$ be Lebesgue measure on$[0,1).$ Prove that $\mu(\phi^{-1}(V)) = \lambda (V).$ we have the following hint:

"It is enough to show this for $V = [\frac{m}{2^n}, \frac{m+1}{2^n})$ for each $n,$ as such sets generate the $\sigma$-algebra for $[0,1).$ what is the preimage of $V$ under $\phi$?"

Here is so far what we know about the setting of the problem:

Consider the probability space $X = \{0,1\}^{\mathbb N}$ of sequences $x = (a_1, a_2, a_3, \dots)$ where each $a_i$ is either $0$ or $1.$ Take $\sigma$ to be the smallest $\sigma$-algebra that contains all cylinder sets $$C(n,A) = \{\textbf{x}| x_n \in A, A \subset \{0,1\}\}$$that is, those sequences whose $n^{th}$ term belongs to some subset $A$ of $\{0,1\}.$ Let $\mu$ be the measure so that, given a finite collection $A_1, \dots , A_n \subset \{0,1\},$ we have$$\mu (\{\textbf{x} \,|\, x_i \in A_i, i = 1, \dots , n \}) = \Pi_{i=1}^{n}\frac{|A_i|}{2}$$Consider the shift map $T: X \to X$ defined by $$T(a_1, a_2, a_3, \dots ) = (a_2, a_3, \dots )$$

Assuming that $T$ is measurable and preserves $\mu.$ We want to show that $T$ is ergodic.

We now know that the set $X_{\infty} = \{ (a_1, a_2, a_3, \dots )\,|\,\exists N s.t. \forall n \geq N, a_n = 1 \}$ has measure zero.

We also, know that the following map is bijective:

$\phi: X\setminus X_{\infty} \to [0,1)$ defined by $$\phi(a_1, a_2, \dots , a_n) = \sum_{n=1}^{\infty}a_n2^{-n}$$ so $\phi$ sends each sequence $(a_1, a_2,\dots)$ to the point $x$ in $[0,1)$ with that binary decimal expansion.

And we may assume that $\phi$ is measurable with measurable inverse.

my questions are:

1- How does this will help us in proving ergodicity?

2- How do we knew that the suggested $V$ in the hint will generate the $\sigma$-algebra for $[0,1)$?

3- Finally, how can the preimage of $V$ under $\phi$ help me in the proof?

Could someone clarify this to me please?