This paper utilizes Wiener's tauberian theorem to indicate that the closed linear span of translations of any simple step function is equal to $L^p[a,b]$, where $1< p \leq \infty$ and $[a,b]$ are real finite constants.
However, I was under the impression that we could use the simple function approximation lemma to approximate any $L^1[a,b]$ function to arbitrary accuracy using translates of simple step functions.
My logic for this would be that the closed linear span of translates of a simple step function contains all simple functions, which are dense in $L^p[a,b]$ for $1\leq p<\infty$ from the simple function approximation lemma.
What is the error in my logic?
Why would the closed linear span of the translates of a single simple step function contain all simple functions? This isn't obvious at all. Consider the simple step function $1_{[0,1]}$. How do you write the characteristic function $1_A$ of an arbitrary measurable set $A$ (of finite measure) as a limit of linear combinations of translates of $1_{[0,1]}$?