Difference between revisions of "Measurable map"

Revision as of 14:23, 16 June 2015 (view source) Alec (Talk | contribs) m ← Older edit		Revision as of 23:40, 2 August 2015 (view source) Alec (Talk | contribs) m Newer edit →
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		+	'''Note: ''' Sometimes called a ''measurable [[Function|fuction]]''<ref name="PAS">Probability and Stochastics - Erhan Cinlar</ref>
		+	__TOC__
	==Definition==		==Definition==
−	Let {{M|(X,\mathcal{A})}} and {{M|(X',\mathcal{A}')}} be [[Measurable space|measurable spaces]] then a map <math>T:X\rightarrow X'</math> is called '''<math>\mathcal{A}/\mathcal{A}'</math>-measurable''', or '''<math>\mathcal{A}-\mathcal{A}'</math>-measurable'''<ref name="PTACC">Probability Theory - A Comprehensive Course - Second Edition - Achim Klenke</ref> if:	+	Let {{M|(X,\mathcal{A})}} and {{M|(X',\mathcal{A}')}} be [[Measurable space|measurable spaces]] then a map:
		+	* <math>T:X\rightarrow X'</math>
		+	is called '''<math>\mathcal{A}/\mathcal{A}'</math>-measurable'''<ref name="MIM">Measures, Integrals and Martingales - Rene Schilling</ref>, or '''<math>\mathcal{A}-\mathcal{A}'</math>-measurable'''<ref name="PTACC">Probability Theory - A Comprehensive Course - Second Edition - Achim Klenke</ref>, or '''Measurable relative to {{M|\mathcal{A} }} and {{M|\mathcal{A}'}}'''<ref name="PAS"/> if:
	* <math>T^{-1}(A')\in\mathcal{A},\ \forall A'\in\mathcal{A}'</math>		* <math>T^{-1}(A')\in\mathcal{A},\ \forall A'\in\mathcal{A}'</math>
−	Note that the function is on {{M|X}} not on {{M|\mathcal{A} }}, I'll write it again to be explicit:
−	* {{M|T:X\rightarrow X'}}
		+
		+	(See also: (Theorem) [[Conditions for a map to be a measurable map]])
	==Notation==		==Notation==
		+	{{Todo|Confirm this - it could just be me getting ahead of myself}}
	A given a [[Measure space|measure space]] (a measurable space equipped with a measure) {{M|(X,\mathcal{A},\mu)}} with a measurable map on the following mean the same thing:		A given a [[Measure space|measure space]] (a measurable space equipped with a measure) {{M|(X,\mathcal{A},\mu)}} with a measurable map on the following mean the same thing:
	* <math>T:(X,\mathcal{A},\mu)\rightarrow(X',\mathcal{A}',\bar{\mu})</math> (if {{M|(X',\mathcal{A}')}} is also equipped with a measure)		* <math>T:(X,\mathcal{A},\mu)\rightarrow(X',\mathcal{A}',\bar{\mu})</math> (if {{M|(X',\mathcal{A}')}} is also equipped with a measure)
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	* [[Probability space]]		* [[Probability space]]
	* [[Measurable space]]		* [[Measurable space]]
−		+	* [[Conditions for a map to be a measurable map]]
	==References==		==References==
	<references/>		<references/>
	{{Definition|Measure Theory}}		{{Definition|Measure Theory}}

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Revision as of 23:40, 2 August 2015

Note: Sometimes called a measurable fuction^[1]

Definition

Let $(X,\mathcal{A})$ and $(X',\mathcal{A}')$ be measurable spaces then a map:

• $$T:X\rightarrow X'$$

is called

$$\mathcal{A}/\mathcal{A}'$$ -measurable^[2], or $$\mathcal{A}-\mathcal{A}'$$ -measurable^[3], or Measurable relative to $\mathcal{A}$ and $\mathcal{A}'$^[1] if:

• $$T^{-1}(A')\in\mathcal{A},\ \forall A'\in\mathcal{A}'$$

(See also: (Theorem) Conditions for a map to be a measurable map)

Notation

---

TODO: Confirm this - it could just be me getting ahead of myself

---

A given a measure space (a measurable space equipped with a measure) $(X,\mathcal{A},\mu)$ with a measurable map on the following mean the same thing:

• $$T:(X,\mathcal{A},\mu)\rightarrow(X',\mathcal{A}',\bar{\mu})$$ (if $(X',\mathcal{A}')$ is also equipped with a measure)
• $$T:(X,\mathcal{A},\mu)\rightarrow(X',\mathcal{A}')$$
• $$T:(X,\mathcal{A})\rightarrow(X',\mathcal{A}')$$

We would write

$$T:(X,\mathcal{A},\mu)\rightarrow(X',\mathcal{A}')$$ simply to remind ourselves of the measure we are using, it is not important to the concept of the measurable map.

As usual, the function is on the first thing in the bracket. (see function for more details)

Motivation

From the topic of random variables - which a special case of measurable maps (where the domain can be equipped with a probability measure, a measure where $X$ has measure 1).

Consider:

$$X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(V,\mathcal{U})$$ , we know that given a $U\in\mathcal{U}$ that $T^{-1}\in\mathcal{A}$ which means we can measure it using $\mathbb{P}$, which is something we'd want to do.

See also

• Random variable
• Probability space
• Measurable space
• Conditions for a map to be a measurable map

References

1. ↑ ^{Jump up to: 1.0 1.1} Probability and Stochastics - Erhan Cinlar
2. Jump up ↑ Measures, Integrals and Martingales - Rene Schilling
3. Jump up ↑ Probability Theory - A Comprehensive Course - Second Edition - Achim Klenke