Simple Function: Difference between revisions
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A measurable function <math>f: X \rightarrow \mathbb{R}</math> is a simple function if <math>f(X)</math> is a finite subset of <math> \mathbb{R} </math>.<ref name="Folland">Folland, Gerald B. (1999). ''Real Analysis: Modern Techniques and Their Applications'', John Wiley and Sons, ISBN 0471317160, Second edition.</ref><ref name="Craig">Craig, Katy. ''MATH 201A Lecture 11''. UC Santa Barbara, Fall 2020.</ref> The standard representation for a simple function is given by | A measurable function <math>f: X \rightarrow \mathbb{R}</math> is a simple function if <math>f(X)</math> is a finite subset of <math> \mathbb{R} </math>.<ref name="Folland">Folland, Gerald B. (1999). ''Real Analysis: Modern Techniques and Their Applications'', John Wiley and Sons, ISBN 0471317160, Second edition.</ref><ref name="Craig">Craig, Katy. ''MATH 201A Lecture 11''. UC Santa Barbara, Fall 2020.</ref> The standard representation for a simple function is given by | ||
<math> f(x) = \sum_{i=1}^n c_i 1_{E_i} (x) </math>, | <math> f(x) = \sum_{i=1}^n c_i 1_{E_i} (x) </math>, | ||
where <math>1_{E_i} (x)</math> is the indicator function on the disjoint sets <math>E_i = f^{-1}(\{c_i\})</math> where <math>f(X) = \{c_1 \dots c_n\}</math>. | where <math>1_{E_i} (x)</math> is the indicator function on the disjoint sets <math>E_i = f^{-1}(\{c_i\})</math> where <math>f(X) = \{c_1, \dots, c_n\}</math>. | ||
==Properties of Simple Functions== | ==Properties of Simple Functions== | ||
Revision as of 23:39, 9 December 2020
The simplest functions you will ever integrate, hence the name.
Definition
A measurable function is a simple function if is a finite subset of .[1][2] The standard representation for a simple function is given by , where is the indicator function on the disjoint sets where .
