From 82d49a75fc70fd35a317c889490822a34de97fa6 Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Mon, 9 Jun 2025 14:45:26 -0500
Subject: [PATCH] fix clarity

---
 pages/Math401/Math401_T1.md  | 9 +++++++++
 pages/Math429/Math429_L17.md | 2 +-
 2 files changed, 10 insertions(+), 1 deletion(-)
 create mode 100644 pages/Math401/Math401_T1.md

diff --git a/pages/Math401/Math401_T1.md b/pages/Math401/Math401_T1.md
new file mode 100644
index 0000000..0611df0
--- /dev/null
+++ b/pages/Math401/Math401_T1.md
@@ -0,0 +1,9 @@
+# Topic 1
+
+## Probability Theory under Language of Measure Theory
+
+### Uniform random numbers
+
+Define picking a random number from the interval $[0,1]$ form the uniform probability distribution.
+
+As a function $f:[0,1]\to S$, where $S$ is the space of potential outcomes of a random phenomenon. (Note, this definition inverts the axis of "probability" and "event" so that we can apply the measure theory to probability theory. Before, we define the probability of an event as a function $P:S\to [0,1]$, where $S\in A$ and $\int_A P(x)dx=1$.)
\ No newline at end of file
diff --git a/pages/Math429/Math429_L17.md b/pages/Math429/Math429_L17.md
index c677e20..22683cf 100644
--- a/pages/Math429/Math429_L17.md
+++ b/pages/Math429/Math429_L17.md
@@ -12,7 +12,7 @@ A **linear functional** on $V$ is a linear map from $V$ to $\mathbb{F}$.
 
 #### Definition 3.110
 
-The **dual space** of V denoted by $V'$ ($\check{V},V^*$) is given by $V'=\mathscr{L}(V,\mathbb{F})$.
+The **dual space** of V denoted by $V'$ (or in some books $\check{V},V^*$) is given by $V'=\mathscr{L}(V,\mathbb{F})$.
 
 The elements of $V'$ are also called **linear functional**.