Quartz sync: Mar 6, 2025, 1:07 PM

content/Definitions/Measure Theory/Lebesgue's Dominated Convergence Theorem.md

@@ -0,0 +1,10 @@
+# Definition
+Let $g$ be a real function on $X$.
+
+Define $g^{+} = \max \{ g, 0 \}$, $g^{-} = -\min \{ g, 0 \}$.
+
+Then $g = g^{+} - g^{-}$ and $g^{\pm} \geq 0$.
+
+> [!example]-
+> ![[Drawing 2025-03-06 11.57.37.excalidraw.dark.svg]]
+%%[[Drawing 2025-03-06 11.57.37.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-06 11.57.37.excalidraw.light.svg|light exported image]]%%

content/Definitions/Measure Theory/Lebesgue's Monotone Convergence Theorem.md

@@ -0,0 +1,4 @@
+# Definition
+Say $X$ has a [[Measure|measure]] $\mu$, and let $f_{n} : X \to [0, \infty]$ be [[Measurable|measurable]] and $f_{1} \leq f_{2} \leq f_{3} \leq \dots$. 
+
+Then $\int f_{m} \, d\mu \to \int \lim_{ n \to \infty } f_{n} \, d\mu$  as $m \to \infty$.