I'm learning Lebesgue integral, where I came to such problem.
Suppose $\varphi$ is a simple function defined on $E$. For each $\varepsilon$, is there a step function $\psi$ approximate to $\varphi$ in the sense that
$|\psi-\varphi|<\varepsilon$ on $E$
or
$\int_E |\psi-\varphi|<\varepsilon$.
Can anyone help me?