Simple Function: Difference between revisions
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==Definition== | ==Definition== | ||
Let <math> (X, \mathcal{M}, \mu) </math> be a measure space. A [[Measurable function | 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} <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 | Let <math> (X, \mathcal{M}, \mu) </math> be a measure space. A [[Measurable function | 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} <ref name="Craig">Craig, Katy. ''MATH 201A Lecture 11''. UC Santa Barbara, Fall 2020.</ref></math>. 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>, | ||
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where <math>1_{E_i} (x)</math> is the indicator function on the disjoint set <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 set <math>E_i = f^{-1}(\{c_i\})</math> where <math>f(X) = \{c_1, \dots, c_n\}</math>. | ||
<ref name="Folland">Folland, Gerald B. (1999). ''Real Analysis: Modern Techniques and Their Applications'', John Wiley and Sons, ISBN 0471317160, Second edition.</ref> | |||
==Properties of Simple Functions== | ==Properties of Simple Functions== | ||
Revision as of 00:06, 10 December 2020
The simplest functions you will ever integrate, hence the name.
Definition
Let be a measure space. A measurable function is a simple function if is a finite subset of Failed to parse (syntax error): {\displaystyle \mathbb{R} <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
,
where is the indicator function on the disjoint set where .
Properties of Simple Functions
Integration of Simple Functions
References
- ↑ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.