Folland real analysis theorem 6.8

Question

On the theorem 6.8, from the real analysis by Folland, it says that simple functions are dense in $L^\infty$. Is this true? For example, a constant function, which never decays belongs to the $L^\infty$ but I cannot see how to approximate this with the simple functions. Also, I found from here (http://math.stanford.edu/~ryzhik/STANFORD/STANF205-16/205_hw3_sol.pdf) that simple functions are not dense in $L[0,1]$. Is this because of the inconsistency of the definition of the simple function? I noticed some require finite measure support.

Answer 1

If $|f(x)| \leq M$ a.e. and $f_n(x)=\sum_i \frac i n I_{E_i}$ where $E_{i}=f^{-1} [\frac {i-1}n , \frac i n)$ then $f_n$ is a simple function and $\|f_n-f||_{\infty}\leq \frac 1n$. Note that the sum in the definition of $f_n$ is actually a finite sum.