PlanetMath: $\sigma$-algebra generated by a random variable

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$\sigma$ -algebra generated by a random variable

(Definition)

Given the probability space $(\Omega,\mathcal{F}, P)$ , any random variable $X\colon \Omega \to \mathbb{R}$ is $\mathcal{F}$ - measurable, in the following sense:

$\displaystyle X^{-1}(U)=\{\omega\in \Omega \colon X(\omega)\in U\} \in \mathcal{F}$

for any open sets $U \subseteq \mathbb{R}$ , or equivalently any Borel sets $U\subset \mathbb{R}$ .

We now define $\mathcal{F}_{X}$ as follows:

$\displaystyle \mathcal{F}_{X}=X^{-1}(\mathcal{B}):=\{X^{-1}(B)\colon B\in \mathcal{B}\},$

where $\mathcal{B}$ is the Borel $\sigma$ -algebra on $\mathbb{R}$ . $\mathcal{F}_X$ is sometimes denoted as $\sigma(X)$ . $\mathcal{F}_{X}$ is a sigma algebra since it satisfies the following:

$\varnothing=X^{-1}(\varnothing)\in \mathcal{F}_{X}$ ,
$\Omega-X^{-1}(B)=X^{-1}(\mathbb{R} - B)\in \mathcal{F}_{X}$ , and
$\bigcup X^{-1}(B_i)=X^{-1}(\bigcup B_i)\in \mathcal{F}_{X}$ .

It is also clear that $\mathcal{F}_X$ is the smallest $\sigma$ -algebra containing all sets of the form $X^{-1}(B)$ , $B\in\mathcal{B}$ . $\mathcal{F}_{X}$ as defined above is called the $\sigma$ -algebra generated by .

" $\sigma$ -algebra generated by a random variable" is owned by PrimeFan. [ full author list (4) | owner history (2) ]

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See Also: $\sigma$ -algebra

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Cross-references: clear, sigma algebra, Borel sets, open sets, random variable, probability space
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This is version 15 of $\sigma$ -algebra generated by a random variable, born on 2006-03-24, modified 2008-06-18.
Object id is 7768, canonical name is MathcalFMeasurableFunction.
Accessed 2530 times total.

Classification:

AMS MSC:	60A99 (Probability theory and stochastic processes :: Foundations of probability theory :: Miscellaneous)
	60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory)

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