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+# 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 $\int_A P(x)dx=1$.)
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@@ -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**.