Pushforward measure

Given two measurable spaces ${\displaystyle (X,{\mathcal {A}})}$ and ${\displaystyle (Y,{\mathcal {B}})}$, a function ${\displaystyle f:X\to Y}$ is called measurable if the pre-image of any measurable set in ${\displaystyle Y}$ is measurable in ${\displaystyle X}$:

${\displaystyle B\in {\mathcal {B}}\implies f^{-1}(B)\in {\mathcal {A}}.}$

If the space ${\displaystyle (X,{\mathcal {A}})}$ is equipped with a measure ${\displaystyle \mu :{\mathcal {A}}\to [0,+\infty ]}$, the pushforward of ${\displaystyle \mu }$ by a measurable function ${\displaystyle f:X\to Y}$ is the measure ${\displaystyle f_{*}(\mu ):{\mathcal {B}}\to [0,+\infty ]}$ defined by

${\displaystyle (f_{*}(\mu ))(B):=\mu \left(f^{-1}(B)\right).}$