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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?

1 Answers 1