Simple function - Revision history

45.42.157.72 at 07:36, 28 January 2025

2025-01-28T07:36:09Z

← Previous revision		Revision as of 07:36, 28 January 2025
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	{{nofootnotes\|date=January 2023}}		{{nofootnotes\|date=January 2023}}
	{{Short description\|Function that attains finitely many values}}		{{Short description\|Function that attains finitely many values}}
	In the [[mathematics\|mathematical]] field of [[real analysis]], a '''simple function''' is a [[real number\|real]] (or [[complex number\|complex]])-valued function over a subset of the [[real line]], similar to a [[step function]]. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be [[measurable function\|measurable]]; as used in practice~~, they invariably are~~.		In the [[mathematics\|mathematical]] field of [[real analysis]], a '''simple function''' is a [[real number\|real]] (or [[complex number\|complex]])-valued function over a subset of the [[real line]], similar to a [[step function]]. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be [[measurable function\|measurable]], as used in practice.

	A basic example of a simple function is the [[floor function]] over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the [[Dirichlet function]] over the real line, which takes the value 1 if ''x'' is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) All [[step function]]s are simple.		A basic example of a simple function is the [[floor function]] over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the [[Dirichlet function]] over the real line, which takes the value 1 if ''x'' is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) All [[step function]]s are simple.

Arthur Walraven: /* Definition */ Use `\colon` in function definition to improve consistency

2025-01-07T09:26:50Z

Definition: Use `\colon` in function definition to improve consistency

← Previous revision		Revision as of 09:26, 7 January 2025
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	==Definition==		==Definition==

	Formally, a simple function is a finite [[linear combination]] of [[indicator function]]s of [[measurable set]]s. More precisely, let (''X'', Σ) be a [[sigma-algebra\|measurable space]]. Let ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> ∈ Σ be a [[sequence]] of disjoint measurable sets, and let ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> be a sequence of [[real number\|real]] or [[complex number]]s. A ''simple function'' is a function <math>f: X \to \mathbb{C}</math> of the form		Formally, a simple function is a finite [[linear combination]] of [[indicator function]]s of [[measurable set]]s. More precisely, let (''X'', Σ) be a [[sigma-algebra\|measurable space]]. Let ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> ∈ Σ be a [[sequence]] of disjoint measurable sets, and let ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> be a sequence of [[real number\|real]] or [[complex number]]s. A ''simple function'' is a function <math>f\colon X \to \mathbb{C}</math> of the form

	:<math>f(x)=\sum_{k=1}^n a_k {\mathbf 1}_{A_k}(x),</math>		:<math>f(x)=\sum_{k=1}^n a_k {\mathbf 1}_{A_k}(x),</math>

Arthur Walraven: /* Integration of simple functions */ Use `\colon` in function definition to improve consistency

2025-01-07T09:25:38Z

Integration of simple functions: Use `\colon` in function definition to improve consistency

← Previous revision		Revision as of 09:25, 7 January 2025
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	==Integration of simple functions==		==Integration of simple functions==

	If a [[measure (mathematics)\|measure]] <math>\mu</math> is defined on the space <math>(X, \Sigma)</math>, the [[Lebesgue integral\|integral]] of a simple function <math>f: X \to \mathbb R</math> with respect to <math>\mu</math> is defined to be		If a [[measure (mathematics)\|measure]] <math>\mu</math> is defined on the space <math>(X, \Sigma)</math>, the [[Lebesgue integral\|integral]] of a simple function <math>f\colon X \to \mathbb R</math> with respect to <math>\mu</math> is defined to be

	:<math>\int_X f d \mu = \sum_{k=1}^na_k\mu(A_k),</math>		:<math>\int_X f d \mu = \sum_{k=1}^na_k\mu(A_k),</math>

JoaoFrancisco1812: /* Integration of simple functions */Added proper latex math typography to a section

2024-09-13T03:30:37Z

Integration of simple functions: Added proper latex math typography to a section

← Previous revision		Revision as of 03:30, 13 September 2024
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	==Integration of simple functions==		==Integration of simple functions==

	If a [[measure (mathematics)\|measure]] μ is defined on the space (''X'',Σ), the [[Lebesgue integral\|integral]] of ''f'' with respect to μ is		If a [[measure (mathematics)\|measure]] <math>\mu</math> is defined on the space <math>(X, \Sigma)</math>, the [[Lebesgue integral\|integral]] of a simple function <math>f: X \to \mathbb R</math> with respect to <math>\mu</math> is defined to be

	:<math>\sum_{k=1}^na_k\mu(A_k),</math>		:<math>\int_X f d \mu = \sum_{k=1}^na_k\mu(A_k),</math>
	if all summands are finite.		if all summands are finite.

LilianaUwU: Reverted edits by Supaaproved (talk) to last version by 2A06:5B01:8201:3600:A18C:3BA3:3EF2:C74A

2024-02-08T18:02:14Z

Reverted edits by Supaaproved (talk) to last version by 2A06:5B01:8201:3600:A18C:3BA3:3EF2:C74A

← Previous revision		Revision as of 18:02, 8 February 2024
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	{{nofootnotes\|date=January 2023}}		{{nofootnotes\|date=January 2023}}
	{{Short description\|Function that attains finitely many values}}		{{Short description\|Function that attains finitely many values}}
	In the [[mathematics\|mathematical]] field of [[real analysis]], '''simple function''' is a [[real number\|real]] (hi [[complex number\|complex]])-valued function over a subset of the [[real line]], similar to a [[step function]]. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be [[measurable function\|measurable]]; as used in practice, they invariably are.		In the [[mathematics\|mathematical]] field of [[real analysis]], a '''simple function''' is a [[real number\|real]] (or [[complex number\|complex]])-valued function over a subset of the [[real line]], similar to a [[step function]]. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be [[measurable function\|measurable]]; as used in practice, they invariably are.

	A basic example of a simple function is the [[floor function]] over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the [[Dirichlet function]] over the real line, which takes the value 1 if ''x'' is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) All [[step function]]s are simple.		A basic example of a simple function is the [[floor function]] over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the [[Dirichlet function]] over the real line, which takes the value 1 if ''x'' is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) All [[step function]]s are simple.
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	[[Bochner measurable function]]		[[Bochner measurable function]]

	== ~~People Who Are Sus~~ ==		== References ==
	*{{aut\|J. F. C. Kingman, S. J. Taylor}}. ''Introduction to Measure and Probability'', 1966, Cambridge.		*{{aut\|J. F. C. Kingman, S. J. Taylor}}. ''Introduction to Measure and Probability'', 1966, Cambridge.
	*{{aut\|[[Serge Lang\|S. Lang]]}}. ''Real and Functional Analysis'', 1993, Springer-Verlag.		*{{aut\|[[Serge Lang\|S. Lang]]}}. ''Real and Functional Analysis'', 1993, Springer-Verlag.

Supaaproved at 17:22, 7 February 2024

2024-02-07T17:22:30Z

← Previous revision		Revision as of 17:22, 7 February 2024
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	[[Bochner measurable function]]		[[Bochner measurable function]]

	== ~~References~~ ==		== People Who Are Sus ==
	*{{aut\|J. F. C. Kingman, S. J. Taylor}}. ''Introduction to Measure and Probability'', 1966, Cambridge.		*{{aut\|J. F. C. Kingman, S. J. Taylor}}. ''Introduction to Measure and Probability'', 1966, Cambridge.
	*{{aut\|[[Serge Lang\|S. Lang]]}}. ''Real and Functional Analysis'', 1993, Springer-Verlag.		*{{aut\|[[Serge Lang\|S. Lang]]}}. ''Real and Functional Analysis'', 1993, Springer-Verlag.

Supaaproved at 17:22, 7 February 2024

2024-02-07T17:22:00Z

← Previous revision		Revision as of 17:22, 7 February 2024
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	{{nofootnotes\|date=January 2023}}		{{nofootnotes\|date=January 2023}}
	{{Short description\|Function that attains finitely many values}}		{{Short description\|Function that attains finitely many values}}
	In the [[mathematics\|mathematical]] field of [[real analysis]], a '''simple function''' is a [[real number\|real]] (or [[complex number\|complex]])-valued function over a subset of the [[real line]], similar to a [[step function]]. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be [[measurable function\|measurable]]; as used in practice, they invariably are.		In the [[mathematics\|mathematical]] field of [[real analysis]], '''simple function''' is a [[real number\|real]] (hi [[complex number\|complex]])-valued function over a subset of the [[real line]], similar to a [[step function]]. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be [[measurable function\|measurable]]; as used in practice, they invariably are.

	A basic example of a simple function is the [[floor function]] over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the [[Dirichlet function]] over the real line, which takes the value 1 if ''x'' is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) All [[step function]]s are simple.		A basic example of a simple function is the [[floor function]] over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the [[Dirichlet function]] over the real line, which takes the value 1 if ''x'' is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) All [[step function]]s are simple.

2A06:5B01:8201:3600:A18C:3BA3:3EF2:C74A: /* See also */

2023-04-10T23:08:35Z

StrokeOfMidnight: Undid revision 1142726630 by Tensorproduct (talk): removed unsourced content, via WP:RS/WP:CITE

2023-03-08T17:40:19Z

Undid revision 1142726630 by Tensorproduct (talk): removed unsourced content, via WP:RS/WP:CITE

← Previous revision		Revision as of 17:40, 8 March 2023
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	where <math>{\mathbf 1}_A</math> is the [[indicator function]] of the set ''A''.		where <math>{\mathbf 1}_A</math> is the [[indicator function]] of the set ''A''.

	=== Countably valued functions ===
	One can extend the definition to an infinite sequence <math>A_1,A_2,\dots,</math> of measurable disjoint sets and an infinite sequence of values <math>a_1,a_2,\dots</math> to define the ''countably valued functions''
	:<math>f(x)=\sum_{k=1}^{\infty} a_k {\mathbf 1}_{A_k}(x).</math>

	==Properties of simple functions==		==Properties of simple functions==
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	converges pointwise to <math>f</math> as <math>n\to\infty</math>. Note that, when <math>f</math> is bounded, the convergence is uniform.		converges pointwise to <math>f</math> as <math>n\to\infty</math>. Note that, when <math>f</math> is bounded, the convergence is uniform.

			==See also==
			[[Bochner measurable function]]

	== References ==		== References ==

Tensorproduct: Undid revision 1142208819 by StrokeOfMidnight (talk)

2023-03-04T01:28:20Z

Undid revision 1142208819 by StrokeOfMidnight (talk)

← Previous revision		Revision as of 01:28, 4 March 2023
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	where <math>{\mathbf 1}_A</math> is the [[indicator function]] of the set ''A''.		where <math>{\mathbf 1}_A</math> is the [[indicator function]] of the set ''A''.

			=== Countably valued functions ===
			One can extend the definition to an infinite sequence <math>A_1,A_2,\dots,</math> of measurable disjoint sets and an infinite sequence of values <math>a_1,a_2,\dots</math> to define the ''countably valued functions''
			:<math>f(x)=\sum_{k=1}^{\infty} a_k {\mathbf 1}_{A_k}(x).</math>

	==Properties of simple functions==		==Properties of simple functions==
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	converges pointwise to <math>f</math> as <math>n\to\infty</math>. Note that, when <math>f</math> is bounded, the convergence is uniform.		converges pointwise to <math>f</math> as <math>n\to\infty</math>. Note that, when <math>f</math> is bounded, the convergence is uniform.

	==See also==
	[[Bochner measurable function]]

	== References ==		== References ==

← Previous revision		Revision as of 23:08, 10 April 2023
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	==See also==		==See also==
	[[Bochner measurable function]]		[[Bochner measurable function]]

	== References ==		== References ==