Rudin's Real and Complex Analysis - up to Section 1.17 review:
From definition 1.13 to Theorem 1.14 and Corollaries should have been just before Fatau's lemma. Both the definition and Theorem are not used till then. So moving on...
Simple Functions:
Simple function is a complex valued function $s$ on a measurable space $X$ whose range consists of finitely many points.
No examples are given. Perhaps, Rudin assumes that the reader is familiar with such functions. And no motivation is given for such an important concept that leads to abstract integration.
Floor function is an example of simple function. Say we have a range of real numbers from $1$ to $10$. Then the floor function yields integers $1,2,\cdots,10$.
For us, better think of simple functions as step functions. Let $A_i=\{x_i:s(x)=\alpha_i\}$ is a partition of space $X$. Clearly $s$ takes a constant value in each $A_i$, $s\|_{A_i}=\alpha_i$. Using characterstic functions, one can write
$$s = \sum_{i=1}^n \alpha_i \chi_{A_i}$$
Measure based integrals are defined using Simple functions. They play similar role as tiny rectangles in the definition of Riemann integrals.
What is very fascinating is that a sequence of non-decreasing simple functions that converge to positive valued measurable function. That is the Theorem 1.17.
One can test this theorem using some measurable functions. If function $f$ is constant, the theorem is true. If the function describes a line through origin (y=x) for positive x axis, then one can define a simple function at each rational point whose pointwise convergence describes the line through origin.
Theorem 1.17: Let $f:X \rightarrow [0,\infty]$ be measurable. There exists simple functions $s_n$ such that
From definition 1.13 to Theorem 1.14 and Corollaries should have been just before Fatau's lemma. Both the definition and Theorem are not used till then. So moving on...
Simple Functions:
Simple function is a complex valued function $s$ on a measurable space $X$ whose range consists of finitely many points.
No examples are given. Perhaps, Rudin assumes that the reader is familiar with such functions. And no motivation is given for such an important concept that leads to abstract integration.
Floor function is an example of simple function. Say we have a range of real numbers from $1$ to $10$. Then the floor function yields integers $1,2,\cdots,10$.
For us, better think of simple functions as step functions. Let $A_i=\{x_i:s(x)=\alpha_i\}$ is a partition of space $X$. Clearly $s$ takes a constant value in each $A_i$, $s\|_{A_i}=\alpha_i$. Using characterstic functions, one can write
$$s = \sum_{i=1}^n \alpha_i \chi_{A_i}$$
Measure based integrals are defined using Simple functions. They play similar role as tiny rectangles in the definition of Riemann integrals.
What is very fascinating is that a sequence of non-decreasing simple functions that converge to positive valued measurable function. That is the Theorem 1.17.
One can test this theorem using some measurable functions. If function $f$ is constant, the theorem is true. If the function describes a line through origin (y=x) for positive x axis, then one can define a simple function at each rational point whose pointwise convergence describes the line through origin.
Theorem 1.17: Let $f:X \rightarrow [0,\infty]$ be measurable. There exists simple functions $s_n$ such that
- $0 \leq s_1 < \leq s_2<\cdots\leq f_n$.
- $s_n(x) \rightarrow f(x)$ as $n \rightarrow \infty$ for every $x \in X$.
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