Which convergence when discussing density in $L^p$?

Question

For an exercise in my measure theory course, I need to prove some density results in the $L^p(\mathbb R^n)$ space. But density requires convergence and this gets me confused. So when discussing density of a subset (e.g. the measurable simple functions) in $L^p$, what convergence are we talking about? Is it pointwise convergence or $L^p$ convergence?

Answer 1

A subset $S$ of $\mathbb L^p\left(X,\mathcal F,\mu\right)$ is dense in $\mathbb L^p\left(X,\mathcal F,\mu\right)$ if for each element $f$ of $\mathbb L^p\left(X,\mathcal F,\mu\right)$ and each positive $\varepsilon$, there exists an element $g$ of $S$ such that $\left\lVert f-g\right\rVert_p:=\left(\int_X\left\lvert f(x)-g(x)\right\rvert^p\mathrm d\mu(x)\right)^{1/p}\lt \varepsilon$, or equivalently, there exists a sequence $\left(g_n\right)_{n\geqslant 1}$ of elements of $S$ such that $\lim_{n\to +\infty}\left\lVert f-g_n\right\rVert_p=0$.