I am trying to solve this problem:
Suppose a function $f$ has a measurable domain and is continuous except at a finite number of points. Is $f$ necessarily measurable?
And I am reading this solution here Is $f$ necessarily measurable? :
"What about this for the first question:
Let $A$ be the measurable domain of $f$ and let $X=\{x_i\}_{i=1}^n$ be the finite collection of discontinuities. Since $|X|$ is finite, we can order them from smallest, say $x_1$, to largest, say $x_n$. Then for $1\le i\le n+1$ define $$ A_i=A\cap (x_{i-1},x_i), $$ where $x_0=-\infty$ and $x_{n+1}=\infty$. Observe that each $A_i$ is measurable and $$ \{x\in A\,:\, f(x)>c\}=\left(\bigcup_{i=1}^{n+1}\{x\in A_i\,:\, f(x)>c\}\right) \cup \{x\in X\,:\,f(x)>c\} $$ For each $c$, each set in the indexed union is measurable (because $f$ is measurable on each A_i) and each subset of $X$ is either empty or contains no more than $n$ points and is therefore measurable as well."
But I have the following question:
Can any finite subset of Lebesgue measurable set be ordered? If so, why?