----- > [!proposition] Proposition. ([[Lp density of simple functions]]) > **1.** Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]] and let $1 \leq p < \infty$. The set $\mathcal{S}^{p}(\mu)$ of all $p$-[[integral|integrable]] [[simple function|simple functions]] is [[dense]] [[norm|in]] $\mathcal{L}^{p}(\mu)$, and [[quotient set|descends]] to a set $S^{p}(\mu)$ [[dense]] in $L^{p}(\mu)$. ^proposition > [!equivalence] > Appealing to the various characterizations of [[dense|density]] that apply in the [[pseudometric|(pseudo)]][[metric space]] setting, the following are equivalent: > 1. $\overline{\mathcal{S}^{p}(\mu)}=\mathcal{L}^{p}(\mu)$ > 2. For all $f \in \mathcal{L}^{p}(\mu)$, there exists a sequence $(s_{n})$ in $\mathcal{S}^{p}(\mu)$ [[converge|converging]] to $f$ > 3. For all $f \in \mathcal{L}^{p}(\mu)$ and all $\varepsilon>0$, there exists $s \in \mathcal{S}^{p}(\mu)$ with $\|f-s\|_{p}<\varepsilon$. > > And similar statements for $S^{p}(\mu)$ and $L^{p}(\mu)$. ^equivalence > [!proof]- Proof. ([[Lp density of simple functions]]) > Fix $f \in \mathcal{L}^{p}$. Per the discussions in [[dense]] (really, [[the sequence lemma]]), it suffices to show there is a [[sequence]] of $p$-[[integral|integrable]] [[simple function|simple functions]] $(f_{k}) \subset \mathcal{S}^{p}$ such that $f_{k} \to f$ in $\mathcal{L}^{p}$. By splitting $f=f^{+}-f^{-}$, it suffices to consider the case that $f$ is nonnegative. > In this case, [[approximation by simple functions]] yields a [[sequence]] $(f_{k}) \subset \mathcal{S}^{p}$ of nonnegative functions such that $(f_{k}) \uparrow f$ in [[pointwise converge|pointwise]] magnitude, i.e., $|f_{k}| \leq |f_{k+1}| \leq |f|$. This gives $|f-f_{k}|^{p} \leq |f|^{p}$ for all $k$[^1] and $|f_{k}-f|^{p} \to 0$, whence the [[Dominated Convergence Theorem]] gives $\lim_{k \to \infty}\|f-f_{k}\|_{p}^{p} = \lim_{k \to \infty} \int |f-f_{k}|^{p} \, d\mu= \int \lim_{k \to \infty} |f_{k}-f|^{p} \, d\mu=0.$ > Next fix $\tilde{f} \in L^{p}$. Lift to $f \in \mathcal{L}^{p}$ and apply the argument above to obtain $(f_{k}) \to f$ in $\mathcal{L}^{p}$ with $f_{k} \in\mathcal{S}^{p}$. Since the [[topological quotient map|quotient projection]] is [[continuous]], $([f_{k}]) \to [f]$ in $L^{p}$. So $S^{p}=[\mathcal{S}^{p}]$ is dense in $L^{p}$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```