Let $f$ be a nonnegative Lebesgue measure function on $\mathbb{R}$, $\epsilon>0$.

How can we approximate $f$ by a nonnegative simple function $s$ with compact support s.t. $s\leq f$ and $$\int|s(x)-f(x)|d\mathcal{L}<\epsilon$$

I know there is sequence of simple functions pointwise converges to $f$, but I am not sure how to find the simple function satisfying the requirement.


Take any sequence of simple functions $s_n$ with $s_n \uparrow f$ and then use $m_n := s_n \cdot \chi_{[-n,n]}$, where $\chi_A$ is the characteristic function (indicator function) of the set $A$.

Then use monotone convergence to conclude that $m_n$ does the trick for $n$ large.


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