From a1b5de688d879069b5e1192057d71572c7bc5368 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Thu, 29 Feb 2024 17:32:24 +0100
Subject: Update

---
 .../lebesgue-integral/convergence-theorems.md      |  6 +++---
 .../lebesgue-integral/the-lp-spaces.md             | 22 ++++++++++++++++++++++
 .../measure-theory/borels-sets.md                  |  2 +-
 .../measure-theory/measurable-maps.md              |  2 +-
 4 files changed, 27 insertions(+), 5 deletions(-)

(limited to 'pages/measure-and-integration')

diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
index 1a34820..6808280 100644
--- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
+++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
@@ -4,9 +4,9 @@ parent: Lebesgue Integral
 grand_parent: Measure and Integration
 nav_order: 2
 description: >
-We state and prove the most important convergence theorems of Lebesgue
-integration theory such as the Monotone Convergence Theorem, Fatou’s Lemma, and the
-Dominated Convergence Theorem.
+  We state and prove the most important convergence theorems of Lebesgue
+  integration theory such as the Monotone Convergence Theorem, Fatou’s Lemma, and the
+  Dominated Convergence Theorem.
 ---
 
 # {{ page.title }}
diff --git a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
index 8482e87..0424117 100644
--- a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
+++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
@@ -23,6 +23,28 @@ Endowed with pointwise addition and scalar multiplication
 $\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space.
 {% endproposition %}
 
+{% proof %}
+We show that $\mathscr{L}^p := \mathscr{L}^p(X,\mathcal{A},\mu)$ is a linear subspace of
+the vector space of all $\KK$-valued functions on $X$.
+The set $\mathscr{L}^p$ is nonempty since
+it contains the zero function.
+Now, suppose $f$ and $g$ are in $\mathscr{L}^p$.
+Then the sum $f+g$ is measurable, because $f$ and $g$ are measurable.
+Moreover, the function $\abs{f+g}^p$ is integrable, because we have the estimate
+
+$$
+\abs{f+g}^p
+\le (\abs{f} + \abs{g})^p
+\le \big\lparen 2 \max(\abs{f},\abs{g}) \big\rparen^p
+\le 2^p (\abs{f}^p + \abs{g}^p),
+$$
+
+where $\abs{f}^p$ and $\abs{g}^p$ are integrable.
+This proves that $f+g$ lies in $\mathscr{L}^p$.
+Finally, it is easy to see that $\alpha f$ lies in $\mathscr{L}^p$
+for any scalar $\alpha \in \KK$.
+{% endproof %}
+
 {% proposition %}
 $\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$.
 {% endproposition %}
diff --git a/pages/measure-and-integration/measure-theory/borels-sets.md b/pages/measure-and-integration/measure-theory/borels-sets.md
index 737a7c8..0cdb142 100644
--- a/pages/measure-and-integration/measure-theory/borels-sets.md
+++ b/pages/measure-and-integration/measure-theory/borels-sets.md
@@ -2,7 +2,7 @@
 title: Borel Sets
 parent: Measure Theory
 grand_parent: Measure and Integration
-nav_order: 2
+nav_order: 3
 ---
 
 # {{ page.title }}
diff --git a/pages/measure-and-integration/measure-theory/measurable-maps.md b/pages/measure-and-integration/measure-theory/measurable-maps.md
index 5b7a76e..dc9f4d7 100644
--- a/pages/measure-and-integration/measure-theory/measurable-maps.md
+++ b/pages/measure-and-integration/measure-theory/measurable-maps.md
@@ -2,7 +2,7 @@
 title: Measurable Maps
 parent: Measure Theory
 grand_parent: Measure and Integration
-nav_order: 3
+nav_order: 2
 ---
 
 # {{ page.title }}
-- 
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