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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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Main: MeasurableFunction