Generated Sigma Algebra

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  1. 1. Generated by a family of sets
  2. 2. Generated by a random variable
  3. 3. Sigma algebras and information

1.  Generated by a family of sets

Given a family \fs4 \mathscr{G} of subsets of the state-space \fs4 \Omega, there is a \fs4 \sigma-algebra that is the smallest one that contains \fs4 \mathscr{G}. This is the \fs4 \sigma-algebra generated by the family of sets \fs4 \mathscr{G} and is denoted by \fs4 \mathscr{F}_{\mathscr{G}} = \mathscr{F}(\mathscr{G}). In particular

\fs4 \mathscr{F}(\mathscr{G}) = \cap \left\{ \mathscr{H} : \mathscr{H} \text{ is a } \sigma\text{-algebra on }\Omega\text{, and }\mathscr{G} \subseteq \mathscr{H} \right\}

2.  Generated by a random variable

Given a random variable \fs4 X(\omega) on the measure space \fs4 (\Omega, \mathcal{F}) there is a \fs4 \sigma-algebra that contains the sets \fs4 X^{-1}(B), for all Borel sets \fs4 B\subset \mathbb{R}. This is the \fs4 \sigma-algebra generated by the random variable \fs4 X and is denoted by

\fs4 \mathscr{F}(X) = \mathscr{F}_X = \left\{ X^{-1}(B) : B \in \mathscr{B} \right\}

3.  Sigma algebras and information

The generated \fs4 \sigma-algebra \fs4 \mathscr{F}_X represents the information we acquire by observing realizations of \fs4 X. Intuitively, knowing that \fs4 X=x allows us to decide in which element of \fs4 \mathscr{F}_X the sample \fs4 \omega belongs.

For two random variables, if \fs4 \mathscr{F}_Y \subseteq \mathscr{F}_X then knowing \fs4 X gives us enough information to determine \fs4 Y. In particular there exists a function \fs4 f such that \fs4 Y = f(X). If \fs4 \mathscr{F}_X !\subseteq \mathcal{F}_Y then this function is not invertible, and \fs4 Y does not determine \fs4 X. If the \fs4 \sigma-algebras are the same, \fs4 \mathscr{F}_Y = \mathscr{F}_X, then the two variables contain exactly the same information: observing one is the same as observing the other.

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