DEFINITION. A function $f$ on $X$ to $\mathbb{R}$ is said to be $\mathcal{A}$ -measurable (or simply measurable) if for every real number $\alpha$ the set
$$ \{x \in X: f(x)>\alpha\} $$ belongs to $\mathcal{A}$.
LEMMA. The following statements are equivalent for a function fon $X$ to $\mathbb{R}$ :
(a) For every $\alpha \in \mathbb{R},$ the set $A_{\alpha}=\{x \in X: f(x)>\alpha\}$ belongs to $\mathcal{A}$
(b) For every $\alpha \in \mathbb{R},$ the set $B_{\alpha}=\{x \in X: f(x) \leqslant \alpha\}$ belongs to $\mathcal{A}$
(c) For every $\alpha \in \mathbb{R},$ the set $C_{\alpha}=\{x \in X: f(x) \geqslant \alpha\}$ belongs to $\mathcal{A}$
(d) For every $\alpha \in \mathbb{R},$ the set $D_{\alpha}=\{x \in X: f(x)<\alpha\}$ belongs to $\mathcal{A}.$
PROOF. Since $B_{\alpha}$ and $A_{\alpha}$ are complements of each other, statement (a) is equivalent to statement (b). Similarly, statements (c) and (d) are equivalent. If (a) holds, then $A_{\alpha-1 / n}$ belongs to $\mathcal{A}$ for each $n$ and since $$ C_{\alpha}=\bigcap_{n=1}^{\infty} A_{\alpha-1 / n} $$ it follows that $C_{\alpha} \in \mathcal{A} . \quad$ Hence (a) implies (c). $\quad$ since $$ A_{\alpha}=\bigcup_{n=1}^{\infty} C_{\alpha+1 / n} $$ it follows that (c) implies (a).
My question is: Why "$C_{\alpha}=\bigcap_{n=1}^{\infty} A_{\alpha-1 / n}$", that is why "$\bigcap$", why not $"\bigcup"$? I didn't understood it, can you explain it? Also I didn't understood that why "$A_{\alpha}=\bigcup_{n=1}^{\infty} C_{\alpha+1 / n}$", that is why "$\bigcup$"?
Let us compare $$B=\bigcup_{n\geq 1}A_{\alpha-1/n}$$ with $$C=\bigcap_{n\geq 1}A_{\alpha-1/n}$$ Which of $B$ and $C$ is the set $C_{\alpha}$, that is $f(x)\geq\alpha$?
If $x\in B$ then $x\in A_{\alpha-1/n}$ for some integer $n$ that is $$f(x)>\alpha-\dfrac{1}{n}$$ for some integer $n$. Then it could very well be that $f(x)<\alpha$, there is some room between $\alpha-1/n$ and $\alpha$.
However if you consider $C$ then $x\in C$ means that $$f(x)>\alpha-\dfrac{1}{n}$$ for every possible integer $n$. That can only happen if $f(x)\geq\alpha$.
The other question is pretty much the same issue, see if you can figure it out on your own.