Quartz sync: Mar 1, 2025, 2:26 PM

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

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+# Definition
+Say $f : X \to Y$ is [[Continuous|continuous]], $\implies f$ is Borel measurable.

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

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+# Definition
+The $\sigma$-algebra generated by the [[Open Sets|open sets]] in a [[Topological Space|topological space]] $X$ is the $\sigma$-algebra of **Borel Sets** of $X$.

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

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+# Definition
+$f: X \to Y$ ([[Topological Space]]) is **measurable** if $f^{-1}(V) \in M$ $\forall \text{ open } V \subset Y$.

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

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+# Definition
+A **measure** on $X$ is a function $\mu : M \to [0,\infty]$ such that:
+1. $\mu(\emptyset) = 0$
+2. $\mu(\cup^{\infty}_{n=1}A_{n}) = \Sigma^{\infty}_{n=1} \mu(A_{n})$ (for pairwise disjoint $A_{n} \in M$)
+Then we say $X$ is a measure space.

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

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+# 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. $A_{n} \in M \implies \cup^{\infty}_{n=1} A_{n} \in M$ ($\implies \cap^{\infty}_{n=1} A_{n} = (\cup^{\infty}_{n=1}A^{\complement})^{\complement} \in M)$)
+# Related Terminologies/Functions
+- [[Measurable]]
+- [[Measure]]