From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/sigma-algebra/index.md | 44 +++++++++++++++---------------
 1 file changed, 22 insertions(+), 22 deletions(-)

(limited to 'source/know/concept/sigma-algebra')

diff --git a/source/know/concept/sigma-algebra/index.md b/source/know/concept/sigma-algebra/index.md
index 5ba7b14..ac607a7 100644
--- a/source/know/concept/sigma-algebra/index.md
+++ b/source/know/concept/sigma-algebra/index.md
@@ -8,15 +8,15 @@ categories:
 layout: "concept"
 ---
 
-In set theory, given a set $\Omega$, a $\sigma$**-algebra**
-is a family $\mathcal{F}$ of subsets of $\Omega$
+In set theory, given a set $$\Omega$$, a $$\sigma$$**-algebra**
+is a family $$\mathcal{F}$$ of subsets of $$\Omega$$
 with these properties:
 
-1.  The full set is included $\Omega \in \mathcal{F}$.
-2.  For all subsets $A$, if $A \in \mathcal{F}$,
-    then its complement $\Omega \!-\! A \in \mathcal{F}$ too.
-3.  If two events $A, B \in \mathcal{F}$,
-    then their union $A \cup B \in \mathcal{F}$ too.
+1.  The full set is included $$\Omega \in \mathcal{F}$$.
+2.  For all subsets $$A$$, if $$A \in \mathcal{F}$$,
+    then its complement $$\Omega \!-\! A \in \mathcal{F}$$ too.
+3.  If two events $$A, B \in \mathcal{F}$$,
+    then their union $$A \cup B \in \mathcal{F}$$ too.
 
 This forms a Boolean algebra:
 property (1) represents TRUE,
@@ -24,28 +24,28 @@ property (1) represents TRUE,
 and that is all we need to define all logic.
 For example, FALSE and OR follow from the above points:
 
-4.  The empty set is included $\varnothing \in \mathcal{F}$.
-5.  If two events $A, B \in \mathcal{F}$,
-    then their intersection $A \cap B \in \mathcal{F}$ too.
+4.  The empty set is included $$\varnothing \in \mathcal{F}$$.
+5.  If two events $$A, B \in \mathcal{F}$$,
+    then their intersection $$A \cap B \in \mathcal{F}$$ too.
 
-For a given $\Omega$, there are typically multiple valid $\mathcal{F}$,
+For a given $$\Omega$$, there are typically multiple valid $$\mathcal{F}$$,
 in which case you need to specify your choice.
-Usually this would be the smallest $\mathcal{F}$
+Usually this would be the smallest $$\mathcal{F}$$
 (i.e. smallest family of subsets)
 that contains all subsets of special interest
 for the topic at hand.
-Likewise, a **sub-$\sigma$-algebra**
-is a sub-family of a certain $\mathcal{F}$,
-which is a valid $\sigma$-algebra in its own right.
-
-A notable $\sigma$-algebra is the **Borel algebra** $\mathcal{B}(\Omega)$,
-which is defined when $\Omega$ is a metric space,
-such as the real numbers $\mathbb{R}$.
-Using that as an example, the Borel algebra $\mathcal{B}(\mathbb{R})$
+Likewise, a **sub-$$\sigma$$-algebra**
+is a sub-family of a certain $$\mathcal{F}$$,
+which is a valid $$\sigma$$-algebra in its own right.
+
+A notable $$\sigma$$-algebra is the **Borel algebra** $$\mathcal{B}(\Omega)$$,
+which is defined when $$\Omega$$ is a metric space,
+such as the real numbers $$\mathbb{R}$$.
+Using that as an example, the Borel algebra $$\mathcal{B}(\mathbb{R})$$
 is defined as the family of all open intervals of the real line,
-and all the subsets of $\mathbb{R}$ obtained by countable sequences
+and all the subsets of $$\mathbb{R}$$ obtained by countable sequences
 of unions and intersections of those intervals.
-The elements of $\mathcal{B}$ are **Borel sets**.
+The elements of $$\mathcal{B}$$ are **Borel sets**.
 
 
 
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