A D18: Map $f : X \to Y$ is

**measurable**with respect to $M_X$ and $M_Y$ if and only if \begin{equation} \sigma_{\text{pullback}} \langle f \rangle \subseteq \mathcal{F}_X \end{equation}

▾ Set of symbols

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Subset

▾ Power set

▾ Hyperpower set sequence

▾ Hyperpower set

▾ Hypersubset

▾ Subset algebra

▾ Subset structure

▾ Measurable space

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Subset

▾ Power set

▾ Hyperpower set sequence

▾ Hyperpower set

▾ Hypersubset

▾ Subset algebra

▾ Subset structure

▾ Measurable space

Formulation 0

Let $M_X = (X, \mathcal{F}_X)$ and $M_Y = (Y, \mathcal{F}_Y)$ each be a D1108: Measurable space.

A D18: Map $f : X \to Y$ is**measurable** with respect to $M_X$ and $M_Y$ if and only if
\begin{equation}
\sigma_{\text{pullback}} \langle f \rangle \subseteq \mathcal{F}_X
\end{equation}

A D18: Map $f : X \to Y$ is

[Comment 0] :
$\sigma_{\text{pullback}} \langle f \rangle$ denotes a D1730: Pullback sigma-algebra with respect to $f$.

Formulation 1

Let $M_X = (X, \mathcal{F}_X)$ and $M_Y = (Y, \mathcal{F}_Y)$ each be a D1108: Measurable space.

A D18: Map $f : X \to Y$ is a**measurable map** from $M_X$ to $M_Y$ if and only if
\begin{equation}
\forall \, E \in \mathcal{F}_Y : f^{-1}(E) \in \mathcal{F}_X
\end{equation}

A D18: Map $f : X \to Y$ is a

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