[[Measurable function]]
# Pushforward measure

Let $(X,\Sigma,\mu)$ be a [[measure space]], $(Y, \mathrm{T})$ be a [[Measure space|measurable space]], and $f: X\to Y$ is a [[measurable function]], then the **pushforward measure** on $Y$ is given by #m/def/measure 
$$
\begin{align*}
f_{*}\mu = \mu \circ (f^{-1} \restriction \mathrm{T})
\end{align*}
$$
i.e. $f_{*}\mu(A) = \mu f^{-1}(A)$.

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