I am currently studyind Density theorems in $L^p$ - spaces. In that, I have encountered a theorem which goes like this -
The space of integrable simple functions is dense in $L^p $(E, $\mathcal{A}$ , $\mu$) where E is a measurable set, $\mathcal{A}$ is the $\sigma -$ algebra and $\mu$ is the measure.
I have some troubles in understanding the denseness of the above given space. Does the denseness mean that union of all the simple function and their limit functions gives me the $L^p$(E, $\mathcal{A}$ , $\mu$) space ? I suppose this denseness is same as that in case of a topological space. Please correct me if I am wrong.
It is the same as a topological space. If $D$ is dense in $L^p$, $$ \forall f \in L^p, \forall \epsilon > 0, \exists \phi \in D\mid \lVert f - \phi \rVert_p = \left(\int |f - \phi|^p\,d\mu\right)^{1/p} < \epsilon $$ The metric is usually the $L^p$ metric unless stated otherwise.