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> | 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}</math>. <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>, | ||
Revision as of 00:07, 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 . [1] The standard representation for a simple function is given by
,
where is the indicator function on the disjoint set where .
