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chore: reorganize all image files and linted documents (#80)
* [PUBLISHER] upload files #62 * [PUBLISHER] upload files #63 * [PUBLISHER] upload files #64 * [PUBLISHER] upload files #65 * [PUBLISHER] upload files #66 * [PUBLISHER] upload files #67 * PUSH NOTE : test-document.md * PUSH NOTE : 09. Lp Functions.md * PUSH ATTACHMENT : mt-09.png * PUSH NOTE : 03. Measure Spaces.md * PUSH ATTACHMENT : mt-03.png * PUSH NOTE : 08. Comparison with the Riemann Integral.md * PUSH ATTACHMENT : mt-08.png * PUSH NOTE : 07. Dominated Convergence Theorem.md * PUSH ATTACHMENT : mt-07.png * PUSH NOTE : 06. Convergence Theorems.md * PUSH ATTACHMENT : mt-06.png * PUSH NOTE : 05. Lebesgue Integration.md * PUSH ATTACHMENT : mt-05.png * PUSH NOTE : test-document.md * [PUBLISHER] upload files #68 * [PUBLISHER] upload files #69 * PUSH NOTE : 08. Comparison with the Riemann Integral.md * PUSH ATTACHMENT : mt-08.png * PUSH NOTE : 02. Construction of Measure.md * PUSH ATTACHMENT : mt-02.png * DELETE FILE : _posts/2023-09-09-test-document.md * DELETE FILE : _posts/Mathematics/Measure Theory/2023-09-09-test-document.md * [PUBLISHER] upload files #70 * PUSH NOTE : Rules of Inference with Coq.md * PUSH NOTE : 수학 공부에 대한 고찰.md * PUSH NOTE : 04. Measurable Functions.md * PUSH ATTACHMENT : mt-04.png * PUSH NOTE : 07. Dominated Convergence Theorem.md * PUSH ATTACHMENT : mt-07.png * PUSH NOTE : 08. Comparison with the Riemann Integral.md * PUSH ATTACHMENT : mt-08.png * PUSH NOTE : 05. Lebesgue Integration.md * PUSH ATTACHMENT : mt-05.png * PUSH NOTE : 03. Measure Spaces.md * PUSH ATTACHMENT : mt-03.png * PUSH NOTE : 09. Lp Functions.md * PUSH ATTACHMENT : mt-09.png * PUSH NOTE : 06. Convergence Theorems.md * PUSH ATTACHMENT : mt-06.png * PUSH NOTE : 02. Construction of Measure.md * PUSH ATTACHMENT : mt-02.png * PUSH NOTE : 01. Algebra of Sets and Set Functions.md * PUSH ATTACHMENT : mt-01.png * PUSH NOTE : 블로그 이주 이야기.md * PUSH ATTACHMENT : blog-logo.png * PUSH ATTACHMENT : github-publisher.png * PUSH NOTE : 05. Services - Enabling Clients to Discover and Talk to Pods.md * PUSH ATTACHMENT : k8s-05.jpeg * PUSH NOTE : 18. Extending Kubernetes.md * PUSH ATTACHMENT : k8s-18.jpeg * PUSH NOTE : 11. Understanding Kubernetes Internals.md * PUSH ATTACHMENT : k8s-11.jpeg * PUSH NOTE : 04. Replication and Other Controllers - Deploying Managed Pods.md * PUSH ATTACHMENT : k8s-04.jpeg * PUSH NOTE : 10. StatefulSets - Deploying Replicated Stateful Applications.md * PUSH ATTACHMENT : k8s-10.jpeg * PUSH NOTE : 02. First Steps with Docker and Kubernetes.md * PUSH ATTACHMENT : k8s-02.jpeg * PUSH NOTE : 06. Volumes - Attaching Disk Storage to Containers.md * PUSH ATTACHMENT : k8s-06.jpeg * PUSH NOTE : 12. Securing the Kubernetes API Server.md * PUSH ATTACHMENT : k8s-12.jpeg * PUSH NOTE : 07. ConfigMaps and Secrets - Configuring Applications.md * PUSH ATTACHMENT : k8s-07.jpeg * PUSH NOTE : 13. Securing Cluster Nodes and the Network.md * PUSH ATTACHMENT : k8s-13.jpeg * PUSH NOTE : 09. Deployments - Updating Applications Declaratively.md * PUSH ATTACHMENT : k8s-09.jpeg * PUSH NOTE : 17. Best Practices for Developing Apps.md * PUSH ATTACHMENT : k8s-17.jpeg * PUSH NOTE : 16. Advanced Scheduling.md * PUSH ATTACHMENT : k8s-16.jpeg * PUSH NOTE : 08. Accessing Pod Metadata and Other Resources from Applications.md * PUSH ATTACHMENT : k8s-08.jpeg * PUSH NOTE : 15. Automatic Scaling of Pods and Cluster Nodes.md * PUSH ATTACHMENT : k8s-15.jpeg * PUSH NOTE : 01. Introducing Kubernetes.md * PUSH ATTACHMENT : k8s-01.jpeg * PUSH NOTE : 03. Pods - Running Containers in Kubernetes.md * PUSH ATTACHMENT : k8s-03.jpeg * PUSH NOTE : 14. Managing Pods' Computational Resources.md * PUSH ATTACHMENT : k8s-14.jpeg * [PUBLISHER] upload files #71 * PUSH NOTE : test-document.md * PUSH NOTE : test-document.md * PUSH ATTACHMENT : test-image.png * DELETE FILE : assets/img/posts/test/test-image.png * [PUBLISHER] upload files #72 * PUSH NOTE : test-document.md * PUSH ATTACHMENT : test-image.png * DELETE FILE : assets/img/posts/test/test-image.png * [PUBLISHER] upload files #73 * PUSH NOTE : test-document.md * PUSH ATTACHMENT : test-image.png * chore: remove test files * [PUBLISHER] upload files #74 * PUSH NOTE : 01. Algebra of Sets and Set Functions.md * PUSH ATTACHMENT : mt-01.png * DELETE FILE : assets/img/posts/Mathematics/Measure Theory/mt-01.png * [PUBLISHER] upload files #76 * PUSH NOTE : 01. Algebra of Sets and Set Functions.md * PUSH ATTACHMENT : mt-01.png * [PUBLISHER] upload files #77 * PUSH NOTE : 09. Lp Functions.md * PUSH ATTACHMENT : mt-09.png * PUSH NOTE : 08. Comparison with the Riemann Integral.md * PUSH ATTACHMENT : mt-08.png * PUSH NOTE : 07. Dominated Convergence Theorem.md * PUSH ATTACHMENT : mt-07.png * PUSH NOTE : 06. Convergence Theorems.md * PUSH ATTACHMENT : mt-06.png * PUSH NOTE : 05. Lebesgue Integration.md * PUSH ATTACHMENT : mt-05.png * PUSH NOTE : 04. Measurable Functions.md * PUSH ATTACHMENT : mt-04.png * PUSH NOTE : 03. Measure Spaces.md * PUSH ATTACHMENT : mt-03.png * PUSH NOTE : 01. Algebra of Sets and Set Functions.md * PUSH ATTACHMENT : mt-01.png * chore: remove images * [PUBLISHER] upload files #78 * PUSH NOTE : 09. Lp Functions.md * PUSH ATTACHMENT : mt-09.png * PUSH NOTE : 08. Comparison with the Riemann Integral.md * PUSH ATTACHMENT : mt-08.png * PUSH NOTE : 07. Dominated Convergence Theorem.md * PUSH ATTACHMENT : mt-07.png * PUSH NOTE : 06. Convergence Theorems.md * PUSH ATTACHMENT : mt-06.png * PUSH NOTE : 05. Lebesgue Integration.md * PUSH ATTACHMENT : mt-05.png * PUSH NOTE : 04. Measurable Functions.md * PUSH ATTACHMENT : mt-04.png * PUSH NOTE : 03. Measure Spaces.md * PUSH ATTACHMENT : mt-03.png * PUSH NOTE : 01. Algebra of Sets and Set Functions.md * PUSH ATTACHMENT : mt-01.png * PUSH NOTE : 18. Extending Kubernetes.md * PUSH ATTACHMENT : k8s-18.jpeg * PUSH NOTE : 17. Best Practices for Developing Apps.md * PUSH ATTACHMENT : k8s-17.jpeg * PUSH NOTE : 16. Advanced Scheduling.md * PUSH ATTACHMENT : k8s-16.jpeg * PUSH NOTE : 15. Automatic Scaling of Pods and Cluster Nodes.md * PUSH ATTACHMENT : k8s-15.jpeg * PUSH NOTE : 14. Managing Pods' Computational Resources.md * PUSH ATTACHMENT : k8s-14.jpeg * PUSH NOTE : 13. Securing Cluster Nodes and the Network.md * PUSH ATTACHMENT : k8s-13.jpeg * PUSH NOTE : 12. Securing the Kubernetes API Server.md * PUSH ATTACHMENT : k8s-12.jpeg * PUSH NOTE : 11. Understanding Kubernetes Internals.md * PUSH ATTACHMENT : k8s-11.jpeg * PUSH NOTE : 10. StatefulSets - Deploying Replicated Stateful Applications.md * PUSH ATTACHMENT : k8s-10.jpeg * PUSH NOTE : 09. Deployments - Updating Applications Declaratively.md * PUSH ATTACHMENT : k8s-09.jpeg * PUSH NOTE : 08. Accessing Pod Metadata and Other Resources from Applications.md * PUSH ATTACHMENT : k8s-08.jpeg * PUSH NOTE : 07. ConfigMaps and Secrets - Configuring Applications.md * PUSH ATTACHMENT : k8s-07.jpeg * PUSH NOTE : 06. Volumes - Attaching Disk Storage to Containers.md * PUSH ATTACHMENT : k8s-06.jpeg * PUSH NOTE : 05. Services - Enabling Clients to Discover and Talk to Pods.md * PUSH ATTACHMENT : k8s-05.jpeg * PUSH NOTE : 04. Replication and Other Controllers - Deploying Managed Pods.md * PUSH ATTACHMENT : k8s-04.jpeg * PUSH NOTE : 03. Pods - Running Containers in Kubernetes.md * PUSH ATTACHMENT : k8s-03.jpeg * PUSH NOTE : 02. First Steps with Docker and Kubernetes.md * PUSH ATTACHMENT : k8s-02.jpeg * PUSH NOTE : 01. Introducing Kubernetes.md * PUSH ATTACHMENT : k8s-01.jpeg * [PUBLISHER] upload files #79 * PUSH NOTE : 02. Construction of Measure.md * PUSH ATTACHMENT : mt-02.png
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@@ -8,10 +8,12 @@ title: "09. $\\mathcal{L}^p$ Functions"
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date: "2023-07-31"
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github_title: "2023-07-31-Lp-functions"
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image:
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path: /assets/img/posts/mt-09.png
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path: /assets/img/posts/Mathematics/Measure Theory/mt-09.png
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attachment:
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folder: assets/img/posts/Mathematics/Measure Theory
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---
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{: .w-50}
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{: .w-50}
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## Integration on Complex Valued Function
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@@ -19,23 +21,23 @@ Let $(X, \mathscr{F}, \mu)$ be a measure space, and $E \in \mathscr{F}$.
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**정의.**
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1. A complex valued function $f = u + iv$, (where $u, v$ are real functions) is measurable if $u$ and $v$ are both measurable.
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1. A complex valued function $f = u + iv$, (where $u, v$ are real functions) is measurable if $u$ and $v$ are both measurable.
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2. For a complex function $f$,
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2. For a complex function $f$,
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$$f \in \mathcal{L}^{1}(E, \mu) \iff \int_E \left\lvert f \right\rvert \,d{\mu} < \infty \iff u, v \in \mathcal{L}^{1}(E, \mu).$$
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3. If $f = u + iv \in \mathcal{L}^{1}(E, \mu)$, we define
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3. If $f = u + iv \in \mathcal{L}^{1}(E, \mu)$, we define
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$$\int_E f \,d{\mu} = \int_E u \,d{\mu} + i\int_E v \,d{\mu}.$$
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**참고.**
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1. Linearity also holds for complex valued functions. For $f_1, f_2 \in \mathcal{L}^{1}(\mu)$ and $\alpha \in \mathbb{C}$,
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1. Linearity also holds for complex valued functions. For $f_1, f_2 \in \mathcal{L}^{1}(\mu)$ and $\alpha \in \mathbb{C}$,
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$$\int_E \left( f_1 + \alpha f_2 \right) \,d{\mu} = \int_E f_1 \,d{\mu} + \alpha \int_E f_2 \,d{\mu}.$$
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2. Choose $c \in \mathbb{C}$ and $\left\lvert c \right\rvert = 1$ such that $\displaystyle c \int_E f \,d{\mu} \geq 0$. This is possible since multiplying by $c$ is equivalent to a rotation.
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2. Choose $c \in \mathbb{C}$ and $\left\lvert c \right\rvert = 1$ such that $\displaystyle c \int_E f \,d{\mu} \geq 0$. This is possible since multiplying by $c$ is equivalent to a rotation.
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Now set $cf = u + vi$ where $u, v$ are real functions and the integral of $v$ over $E$ is $0$. Then,
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@@ -109,9 +111,9 @@ We treat $[f]$ as an element in $\mathcal{L}^{p}(X, \mu)$, and write $f = [f]$.
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**참고.**
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1. We write $\left\lVert f \right\rVert_p = 0 \iff f = [0] = 0$ in the sense that $f = 0$ $\mu$-a.e.
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1. We write $\left\lVert f \right\rVert_p = 0 \iff f = [0] = 0$ in the sense that $f = 0$ $\mu$-a.e.
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2. Now $\lVert \cdot \rVert_p$ is a **norm** in $\mathcal{L}^{p}(X, \mu)$ so $d(f, g) = \left\lVert f - g \right\rVert_p$ is a **metric** in $\mathcal{L}^{p}(X, \mu)$.
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2. Now $\lVert \cdot \rVert_p$ is a **norm** in $\mathcal{L}^{p}(X, \mu)$ so $d(f, g) = \left\lVert f - g \right\rVert_p$ is a **metric** in $\mathcal{L}^{p}(X, \mu)$.
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## Completeness of $\mathcal{L}^p$
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@@ -119,9 +121,9 @@ Now we have a *function space*, so we are interested in its *completeness*.
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**정의.** (Convergence in $\mathcal{L}^p$) Let $f, f_n \in \mathcal{L}^{p}(\mu)$.
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1. $f_n \rightarrow f$ in $\mathcal{L}^p(\mu) \iff \left\lVert f_n-f \right\rVert_p \rightarrow 0$ as $n \rightarrow\infty$.
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1. $f_n \rightarrow f$ in $\mathcal{L}^p(\mu) \iff \left\lVert f_n-f \right\rVert_p \rightarrow 0$ as $n \rightarrow\infty$.
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2. $\left( f_n \right)_{n=1}^\infty$ is a Cauchy sequence in $\mathcal{L}^{p}(\mu)$ if and only if
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2. $\left( f_n \right)_{n=1}^\infty$ is a Cauchy sequence in $\mathcal{L}^{p}(\mu)$ if and only if
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> $\forall \epsilon > 0$, $\exists\,N > 0$ such that $n, m \geq N \implies \left\lVert f_n-f_m \right\rVert_p < \epsilon$.
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@@ -206,4 +208,3 @@ $$\left\lVert f - \sum_{k=1}^{m} a_k g_n^k \right\rVert_p = \left\lVert \sum_{k=
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Next for $f \in \mathcal{L}^{p}$ and $f \geq 0$, there exist simple functions $f_n \geq 0$ such that $f_n \nearrow f$ in $\mathcal{L}^{p}$. Finally, any $f \in \mathcal{L}^{p}$ can be written as $f = f^+ - f^-$, which completes the proof.
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이러한 확장을 몇 번 해보면 굉장히 routine합니다. $\chi_F$ for closed $F$ $\rightarrow$ $\chi_A$ for measurable $A$ $\rightarrow$ measurable simple $f$ $\rightarrow$ $0\leq f \in \mathcal{L}^{p} \rightarrow$ $f \in \mathcal{L}^{p}$ 와 같은 순서로 확장합니다.
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