Preliminaries

As usual, a step function $\eta$ is a function that has only finitely many values and is of the form: $$\eta(x) = \begin{cases} \eta_1,\quad x\in[a=t_0, t_1]\\ \eta_2,\quad x\in(t_1, t_2]\\ ...\\ \eta_n,\quad x\in(t_{n-1}, b = t_n] \end{cases}$$ We say that a function $f$ is $L_1$-approximable on $\Omega$ by a step functions $\eta$ iff $$\forall \epsilon>0,\quad \exists \eta_\epsilon\in \{\eta_n\}: \int_\Omega |f(x) - \eta_n(x)|dx < \epsilon,$$ where $\{\eta_n\}$ is the set of all step functions.

Question

Is every real function $L_1$-approximable by a step functions? If not, what conditions must be satisfied for those functions which is approximable in the latter sense?

Modification

Maybe it is interesting to modify definition of a step function and allow it to assume infinitely but countably many values.

• Yes, ever $L^1$ function can be approximated in this way. For giving you a proof, it would be good to know what you already know about integrable functions. Do you know that they can be approximated by continuous functions with compact support? – PhoemueX Nov 13 '17 at 8:01