Suppose $f$ is a Lebesgue measurable function.
I know that if $f$ is non-negative then we can approximate $f$ by a sequence of simple functions.
But I came across the statement " every Lebesgue measurable function is equal almost everywhere to a limit of simple Lebesgue measurable functions".
I'm not sure why this holds if $f$ is negative.
Thanks!
Best Answer
We have that $f = f^+ - f^-$ where $f^+ = \max{f, 0}$ and $f^-=\max{-f,0}$. Note that since $f$ is measurable, we have that $f^+$ and $f^-$ are measurable.
Let $\{h_n\}_n$ be a sequence of simple function converging almost everywhere to $f^+$ and let $\{k_n\}_n$ be a sequence of simple function converging almost everywhere to $f^-$. It folows that $\{h_n-k_n\}_n$ is a a sequence of simple function converging almost everywhere to $f^+- f^-$, that means, converging almost everywhere to $f$.