ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D1732
Pushforward measure
Formulation 0
Let $M_X = (X, \mathcal{F}_X, \mu)$ be a D1158: Measure space.
Let $M_Y = (Y, \mathcal{F}_Y)$ be a D1108: Measurable space.
Let $f : X \to Y$ be a D201: Measurable map from $M_X$ to $M_Y$.
The pushforward measure on $M_Y$ with respect to $f$ and $M_X$ is the D4361: Unsigned basic function \begin{equation} \mathcal{F}_Y \to [0, \infty], \quad E \mapsto \mu[f^{-1}(E)] \end{equation}
Formulation 1
Let $M_X = (X, \mathcal{F}_X, \mu)$ be a D1158: Measure space.
Let $M_Y = (Y, \mathcal{F}_Y)$ be a D1108: Measurable space.
Let $f : X \to Y$ be a D201: Measurable map from $M_X$ to $M_Y$.
The pushforward measure on $M_Y$ with respect to $f$ and $M_X$ is the D4361: Unsigned basic function \begin{equation} \mathcal{F}_Y \to [0, \infty], \quad E \mapsto \mu(f^{-1} E) \end{equation}
Results
R4609
R4610
R4611
Pushforward change of variables formula for unsigned basic integral