induced measure

definition

For a measure space $(X,\mathcal{S},\mu )$ and a 20231205121417-sigma_algebra $(Y,\mathcal{F})$, with a measurable function/ map $f:X\to Y$ the induced measure is

$$\mu \circ f^{-1}(U)$$

Proof this is a measure

1. $\mu (E)\ge 0\forall E\in \mathcal{S}⟹\mu (f^{-1}(F))\ge 0\forall F\in \mathcal{F}$
2. $\mu (f^{-1}(\varnothing ))=0$ because $f^{-1}(\varnothing )=\varnothing$
3. Countable additivity holds because for $E_1$ and $E_2$ disjoint, $f^{-1}(E_1)$ and $f^{-1}(E_2)$ are disjoint.