Let $(X, \mathscr A, \mu)$ be $\sigma$ - algebra and let's define function $f:X\rightarrow Y$. Let's define a family of sets $$\mathscr B=\{B\subset Y:f^{-1}(B)\in\mathscr A\}$$and function $$\nu: \mathscr B\rightarrow\nu(B)=\mu(f^{-1}(B))\in[0,+\infty).$$ Check if $\mathscr B$ is also a $\sigma$-algebra on a set $Y$ and if $\nu$ is a measure.
I checked the first property, which is that an empty set is in $\mathscr B$. I managed to prove the second property as well, but only under the condition that $f$ is bijective, but I'm not really sure what about the third one.