Image dimension

A picture measure is a term from the mathematical branch of measure theory and is used to transfer the measure in one measure space to another room . Here, values ​​are assigned to the quantities with the help of a measurable function . The dimension so defined is the image dimension. ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle (\Omega ',\Sigma ')}$ ${\displaystyle g\colon \Omega \to \Omega '}$ ${\displaystyle \Sigma '}$ ${\displaystyle \Omega '}$

The image size plays an important role, especially when defining the distribution of a random variable .

definition

It is a measure space and one - measurable function in a measuring room . Then ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle g\colon \Omega \to \Omega '}$ ${\displaystyle \Sigma {\text{-}}\Sigma '}$ ${\displaystyle (\Omega ',\Sigma ')}$

${\displaystyle \mu ':=\mu \circ g^{-1}\colon \Sigma '\to [0,\infty ],\quad \Sigma '\ni A'\mapsto \mu (g^{-1}(A'))\in [0,\infty ]}$

a measure of , the image measure of in relation to . Here referred to the archetype of . ${\displaystyle (\Omega ',\Sigma ')}$ ${\displaystyle \mu '}$ ${\displaystyle \mu }$ ${\displaystyle g}$ ${\displaystyle g^{-1}(A')}$ ${\displaystyle A'\in \Sigma '}$

Transformation set

For a measurable function (where the (affine) extended real numbers denote) the following transformation theorem applies to measurable sets : ${\displaystyle f\colon \Omega '\to {\overline {\mathbb {R} }}}$ ${\displaystyle {\overline {\mathbb {R} }}:=\mathbb {R} \cup \{-\infty ,+\infty \}}$ ${\displaystyle A\subseteq \Omega '\;}$

${\displaystyle \int _{g^{-1}(A)}f\circ g\;\mathrm {d} \mu =\int _{A}f\;\mathrm {d} (\mu \circ g^{-1})}$,

if at least one of the above two integrals is defined.

swell

1. ^ Robert B. Ash: Real Analysis and Probability. Academic Press, New York 1972. ISBN 0-12-065201-3 . Theorem 1.6.12.