I am self-studying Measure Theory by Donald Cohn. When proving this statement:
If $f\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$, then $\{x\in X:|f(x)|>\|f\|_{\infty}\}$ is locally $\mu$-null.
the book wrote:
If $\{M_n\}$ is a nonincreasing sequence of real numbers such that $\|f\|_{\infty}=\lim_{n\to\infty}M_n$ and such that for each $n$ the set $\{x\in X:|f(x)|>M_n\}$ is locally $\mu$-null, then the set $\{x\in X:|f(x)|>\|f\|_{\infty}\}$ is the union of the sets $\{x\in X:|f(x)|>M_n\}$ and so is locally $\mu$-null.
I can see that the union of a sequence of locally $\mu$-null sets is locally $\mu$-null. But my question is: How can we make sure that such a sequence $\mathbf{\{M_n\}}$ exists? Could someone please explain this for me? Thank you very much!
Some related definitions:
Definition$\quad$ A subset $B$ of $X$ is $\mu$-negligible (or $\mu$-null) if there is a subset $A$ of $X$ such that $A\in\mathscr{A}$, $B \subseteq A$, and $\mu(A)=0$.
Definition$\quad$ Let $\mathscr{L}^{\infty}(X,\mathscr{A},\mu,\mathbb{R})$ be the set of all bounded real-valued $\mathscr{A}$-measurable functions on $X$, and let $\mathscr{L}^{\infty}(X,\mathscr{A},\mu,\mathbb{C})$ be the set of all bounded complex-valued $\mathscr{A}$-measurable functions on $X$.
In discussions that are valid for both real- and complex-valued functions we will often use $\mathscr{L}^p(X,\mathscr{A},\mu)$ to represent either $\mathscr{L}^p(X,\mathscr{A},\mu,\mathbb{R})$ or $\mathscr{L}^p(X,\mathscr{A},\mu,\mathbb{C})$.
Definition$\quad$ A subset $N$ of $X$ is locally $\mu$-null if for each set $A$ that belongs to $\mathscr{A}$ and satisfies $\mu(A)<+\infty$ the set $A\bigcap N$ is $\mu$-null. A property of points of $X$ is said to hold \textit{locally almost everywhere} if the set of points at which it fails to hold is locally null.
Definition$\quad$ We can define $\|\cdot\|_p$ in the case where $p=+\infty$ by letting $\|f\|_{\infty}$ be the infimum of those nonnegative numbers $M$ such that $\{x\in X:|f(x)|>M\}$ is locally $\mu$-null.