From 7c66b227a494748e2a546fb85317accd00aebe53 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Thu, 15 Feb 2024 05:11:07 +0100
Subject: Update

---
 .../lebesgue-integral/convergence-theorems.md                       | 6 +++---
 pages/measure-and-integration/lebesgue-integral/index.md            | 2 +-
 pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md    | 2 +-
 pages/measure-and-integration/measure-theory/measures.md            | 2 +-
 pages/measure-and-integration/measure-theory/sigma-algebras.md      | 1 -
 pages/measure-and-integration/measure-theory/signed-measures.md     | 2 +-
 6 files changed, 7 insertions(+), 8 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 67f0996..f9ebc4a 100644
--- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
+++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
@@ -32,10 +32,10 @@ $$
 In the following proof we omit $X$ and $d\mu$ for visual clarity.
 
 {% proof %}
-By definition, we have $\liminf_{n \to \infty} f_n = \lim_{n \to \infty} g_n$, where $g_n = \inf_{k \ge n} f_k$.
+By definition, we have $\liminf_{n \to \infty} f_n = \lim_{n \to \infty} g_n$,
+where $g_n = \inf_{k \ge n} f_k$.
 Now $(g_n)$ is a monotonic sequence of nonnegative measurable functions.
-By the
-[Monotone Convergence Theorem](#monotone-convergence-theorem)
+By the [Monotone Convergence Theorem](#monotone-convergence-theorem)
 
 $$
 \int \liminf_{n \to \infty} f_n = \lim_{n \to \infty} \int g_n.
diff --git a/pages/measure-and-integration/lebesgue-integral/index.md b/pages/measure-and-integration/lebesgue-integral/index.md
index a857d95..3418e10 100644
--- a/pages/measure-and-integration/lebesgue-integral/index.md
+++ b/pages/measure-and-integration/lebesgue-integral/index.md
@@ -106,7 +106,7 @@ For any measurable subset $A \subset X$ we define
 the *integral on $A$* of a (quasi-)integrable function $f : X \to \overline{\RR}$ (or $\CC$) by
 
 $$
-\int_A f \, d\mu = 
+\int_A f \, d\mu =
 \int_X \chi_A f \, d\mu.
 $$
 {% enddefinition %}
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 023c253..8482e87 100644
--- a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
+++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
@@ -1,5 +1,5 @@
 ---
-title: The L<sup>p</sup> Spaces 
+title: The L<sup>p</sup> Spaces
 parent: Lebesgue Integral
 grand_parent: Measure and Integration
 nav_order: 4
diff --git a/pages/measure-and-integration/measure-theory/measures.md b/pages/measure-and-integration/measure-theory/measures.md
index 637ab0c..c843881 100644
--- a/pages/measure-and-integration/measure-theory/measures.md
+++ b/pages/measure-and-integration/measure-theory/measures.md
@@ -14,7 +14,7 @@ is mapping $\mu : \mathcal{A} \to [0,\infty]$ such that
 - $\mu(\varnothing) = 0$,
 - for every sequence $(A_n)_{n \in \NN}$ of
   pairwise disjoint sets $A_n \in \mathcal{A}$
-  
+
   $$
   \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen
   = \sum_{n=0}^{\infty} \mu(A_n).
diff --git a/pages/measure-and-integration/measure-theory/sigma-algebras.md b/pages/measure-and-integration/measure-theory/sigma-algebras.md
index 5d22f6b..8f58f09 100644
--- a/pages/measure-and-integration/measure-theory/sigma-algebras.md
+++ b/pages/measure-and-integration/measure-theory/sigma-algebras.md
@@ -47,4 +47,3 @@ defined to be the intersection of all σ-algebras on $X$ containing $\mathcal{A}
 By the previous proposition, $\sigma(\mathcal{E})$ is in fact a σ-algebra on $X$.
 
 ## Products of {{ page.title }}
-
diff --git a/pages/measure-and-integration/measure-theory/signed-measures.md b/pages/measure-and-integration/measure-theory/signed-measures.md
index 77b2416..657a28f 100644
--- a/pages/measure-and-integration/measure-theory/signed-measures.md
+++ b/pages/measure-and-integration/measure-theory/signed-measures.md
@@ -18,7 +18,7 @@ is a mapping $\mu : \mathcal{A} \to [-\infty,\infty]$ such that
 - for every sequence $(A_n)_{n \in \NN}$ of
   pairwise disjoint sets $A_n \in \mathcal{A}$
   {: .my-0 }
-  
+
   $$
   \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen
   = \sum_{n=0}^{\infty} \mu(A_n).
-- 
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