Dense chain in $L^p(E)$ space

For dense chain I mean chain like the following form (roughly)[denote dense:=$\subset\subset$]:

Polynomials with rational coefficients $\subset\subset$ Polynomials $\subset\subset$ Continuous function with compact support $\subset\subset$ Continuous functions $\subset\subset$Step function $\subset\subset$ Simple functions $\subset\subset$ $L^p$

Can anyone help me make this chain more precise? Thanks

Some definitions:

step function $f=\sum_{k=1}^nc_k\chi_{I_k},x\in [a,b],\sqcup_{k=1}^nI_k=[a,b]$, where $I_k$ is interval.

Simple function $f=\sum_{k=1}^nc_k\chi_{E_k},x\in [a,b],\sqcup_{k=1}^nE_k=E$, where $E_k$ is measurable subset of measurable set $E$.

• What are your definitions of step functions and simple functions? And generally, simple functions are not continuous, and continuous functions are not simple functions. – Hetebrij May 12 '16 at 23:38

Several of these inclusions are incorrect. Polynomials don't have compact support (and generally won't be in $L^p$ unless $E$ is a bounded subset of $\mathbb{R}^n$), and continuous functions are not step functions. And again, not every continuous function will be in $L^p$, depending on what $E$ is.
Another important dense subset of $L^p(\mathbb{R}^n)$ (for $1\leq p<\infty$) is the set of infinitely differentiable functions with compact support. It's far from obvious that this set of functions is dense in $L^p$, but it can be proved using convolutions.