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--- math105-s22:notes:lecture_9 [2022/02/15 04:55]
pzhou created
+++ math105-s22:notes:lecture_9 [2026/02/21 14:41] (current)
@@ Line 20: / Line 20: @@
 Simple functions forms a vector space (i.e., closed under addition and scalar multiplication), and can be written as a finite linear combination of characteristic functions $\chi_E$.
-The important thing is that, any **non-negative** measurable function $f$ admits a sequence of simple functions $f_n$, non-negative, and $f_n \leq f_{n+1}$,  such that $f_n \to f$ pointwise.
+The important thing is that, any **non-negative** measurable function $f$ admits a sequence of simple functions $f_n$, non-negative, and $f_n \leq f_{n+1}$,  such that $f_n \to f$ pointwise. The construction requires both 'trunction' and refinement.
+We then define integration for simple functions. Integration is a linear map from the vector space of simple function to $\R$.
+===== 8.2 Integration for non-negative functions =====
+Finally, in 8.2, we will define integration for non-negative measurable functions.
+$\int f = \sup \{ \int s \mid 0 \leq s \leq f, \text{$s$ is a simple function  \} $
+For $f,g : \Omega \to [0, \infty]$, how to prove $\int f+g = \int f + \int g$?
-We then define

Lecture Notes