Properties of Lebesgue functions

Question

If $f\in \mathcal {L}$ then there exists a sequence $\{f_k\}$ of step functions s.t. $\lim_{k\to\infty} f_k(x)=f(x)$ for almost all $x$ and $$\lim_{k\to\infty} \int|f(x)-f_k(x)|\,dx=0.$$ If I have shown the first step how do I show the second. Thank you for your help.

Answer 1

Fix $\dot f$ an equivalence class for equality almost everywhere and choose $f$ in this class, a measurable function. Writing $f=\max\{0,f\}+(f-\max\{0,f\})$, we write $f$ as the difference of two non-negative measurable and integrable functions. So it's enough to deal with the case $f$ non-negative.

Approximate pointwise $f$ by a non-decreasing sequence $\{f_k\}$ of step function. The fact that $\int |f-f_k|\to 0$ follows from monotone convergence theorem.