example of a prop of measurable functions

Question

We proved in class that a function $f$ is lebesgue measurable if there exists a increasing sequence of simple functions that converge pointwise to $f$. That's OK but how can I build this sequence given the function? For example how could be a increasing sequence of simple functions that converge pointwise to the identity function?

Answer 1

For example, if you want to approximate $f$, you could let $f_n(x)$ be the largest rational number $p/q$ such that $|p|+|q|<n$ and $p/q\le f(x)$, or $0$ if no such rational exists.

(For any given $n$ there are only finitely many $p/q$ that satisfy the first condition, so $f_n$ is necessarily simple).