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Good evening,

I'm having a little trouble with a problem, and I was wondering if you guys could help me out. I'm just not sure what to use and I feel like I am given such little information.

So first, we are given that $f \in L^1(\Omega,A,\mu)$. Now I have to prove that for any $\epsilon > 0 $ you can find a bounded function $g$ with the property that $\int_{\Omega} |f-g|d \mu . Also, prove there is a simple function $s$ with the same property.

So really all I know is that $\int_{\Omega} |f|d \mu < \infty$, because $f \in L^1(\Omega,A,\mu)$. There has to be a clever way to construct the $g$ and $s$, but I'm not seeing it.

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Put $f_n(x) = f(x)$ if $|f(x)|\le n$, $f_n(x) = n$ if $f(x) > n$ and $-n$ if $f_n(x) = f(x) < -n$. These functions are bounded. A simple monotone convergence argument shows that $f_n\to f$ in the $L^1$ sense.

Now finding the simple function will not prove difficult.