----- > [!proposition] Proposition. ([[antimonotone inclusion of Lebesgue spaces wrt a finite measure]]) > Suppose $(X, \Sigma, \mu)$ is a finite [[measure|measure space]] and $0<p<q<\infty$. Then $\|f\|_{p} \leq \mu(X)^{\frac{1}{p}-\frac{1}{q}} \|f\|_{q}$ for all $f \in \mathcal{L}^{q}(\mu)$. It follows that $\mathcal{L}^{q}(\mu) \subset \mathcal{L}^{p}(\mu)$, and that convergence in $q$-norm implies convergence in $p$-norm. ^proposition > [!proof]- Proof. ([[antimonotone inclusion of Lebesgue spaces wrt a finite measure]]) > Fix $f \in \mathcal{L}^{q}(\mu)$. Let $r=\frac{q}{p}$, [[dual exponent|so that]] $r'=\frac{q}{q-p}$. Note $r>1$. By [[Hölder's inequality]][^1], $\begin{align} \int |f|^{p} \, d\mu &\leq \left( \int |f|^{pr} \, d\mu \right)^{1/r}\left( \int |1|^{pr'} \, d\mu \right)^{1/r'} \\ &= \mu(X)^{(q-p)/pq} \left( \int |f|^{q} \, d\mu \right)^{1/r}. \end{align}$ Now raise both sides of the inequality to the power $\frac{1}{p}$, getting $\left( \int |f|^{p} \, d\mu \right)^{1/p} \leq \mu(X)^{(q-p)/(pq)} \left( \int |f|^{q} \, d\mu \right)^{1/q}$ which is the desired inequality. [^1]: Namely, [[Hölder's inequality]] with $p$ replaced by $r$ and $f$ replaced by $|f|^{p}$ and $h$ replaced by $1$. ----- #### ---- #### 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 > ```