Quartz sync: Mar 3, 2025, 12:08 PM

content/Definitions/Measure Theory/Sigma-Algebra/Borel Sigma-Algebra.md

@@ -0,0 +1 @@
+**Borel $\sigma$-algebra** on a [[Topological Space|topological space]] $X$ is the one generated by $\tau$.

content/Definitions/Measure Theory/Sigma-Algebra/Sigma-Algebra.md

@@ -1,8 +1,8 @@
 # Definition
 A $\sigma$-algebra in a set $X$ is a collection of subsets, so called measurable sets, of $X$ such that (the requirements are):
 1. $X \in M$
-2. $A \in M \implies A^{\complement} \in M$ ($X^{\complement} = \emptyset \in M$)
-3. $A_{n} \in M \implies \cup^{\infty}_{n=1} A_{n} \in M$ ($\implies \cap^{\infty}_{n=1} A_{n} = (\cup^{\infty}_{n=1}A^{\complement})^{\complement} \in M)$)
+2. $A \in M \implies A^{\complement} \in M$   ($A^{\complement} \equiv X \setminus A$) ($X^{\complement} = \emptyset \in M$)
+3. $A_{n} \in M \implies \cup^{\infty}_{n=1} A_{n} \in M$   ($\implies \cap^{\infty}_{n=1} A_{n} = (\cup^{\infty}_{n=1}A^{\complement})^{\complement} \in M)$)
 # Related Terminologies/Functions
 - [[Measurable]]
 - [[Measure]]

content/Definitions/Measure Theory/Simple Function.md

@@ -0,0 +1,10 @@
+A **simple function** on $X$ is a function $s : X \to \mathbb{R}$ of the form $s = \Sigma_{i=1}^{n} a_{i} \times X_{a_{i}}$ for pairwise disjoint $A_{i} \subset X$ and distinct real numbers $a_{i}$
+
+$X_{A}(x) = \begin{cases}1 & \text{If}\ x \in A\\ 0 & \text{If}\ x \notin A\end{cases}$
+
+> [!note]
+> If $X$ has $M$ then: $s$ is [[Measurable|measurable]] $\iff$ all $A_{i}$ are [[Measurable|measurable]]
+> 
+> $A_{i} = s^{-1}(\{ a_{i} \}) = s^{-1}(\mathbb{R} \setminus \{ a_{i} \})^{\complement}$
+
+If we have a [[Measure|measure]] $\mu$ on $X$, define $\int_{A} s \, d\mu \equiv \Sigma_{i=1}^{n} a_{i} \times \mu (A \cap A_{i})$ for $A \in M$  (they are all $\in [0, \infty])$.