Measurable Function

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November 03, 2006, at 11:23 AM by Kyriakos -
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Consider a function \fs4 f which maps between two measure spaces \fs4 (\Omega, \cal{F}) and \fs4 (\Psi, \cal{G})

\fs4 f : \Omega \rightarrow \Psi

The function \fs4 f is \fs4 (\cal{F}, \cal{G})-measurable if

for all \fs4 G \in \cal{G} we have \fs4 f^{-1}(G) \in \cal{F}

Measurable functions are very general constructs, and will include most functions we encounter. Non-measurable functions are considered pathological.

If \fs4 \Psi = \mathbb{R}^n and \fs4 \cal{G}=\cal{B}(\mathbb{R}^n) is the corresponding Borel algebra then we will call the function just \fs4 \cal{F}-measurable. In probability theory a random variable is such a function.

If \fs4 \Omega = \mathbb{R}^n and \fs4 \cal{F}=\cal{B}(\mathbb{R}^n) as well, then we will call the function just measurable. For example continuous functions are measurable.

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