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--- math105-s22:notes:lecture_9 [2022/02/14 20:55]
pzhou created
+++ math105-s22:notes:lecture_9 [2022/02/14 21:41] (current)
pzhou [8.1 Simple function]
@@ 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