I need some help with subparagraph b) of the question.
$X$ and $Y$ are sets and $f$ a mapping from $X$ to $Y$. Let $\mathcal{A}$ be a sigma-algebra on X.
A) Show that the collection $\mathcal{B} = \{ B\subseteq Y: f^{-1}(B)\in \mathcal{A} \}$ is a $\sigma$- algebra on $Y$.
b) Let $C$ be the $\sigma$-algebra generated in $Y$ by
$f(\mathcal{A}) =\{ f (A): A \in \mathcal{A} \} $. Show that if f is injective, then $C \subseteq \mathcal{B}$ and if $f$ is surjective, then $\mathcal{B} \subseteq C$.
I dont know how work with the set generated by $f(\mathcal{A})$ and the inyectivity, surjectivity of the preimage… THX
Best Answer
Here are two general results you can use. Let $f:X\to Y$ be any function.
Let $S\subseteq X$. If $f$ is injective, then $S=f^{-1}\big(f(S)\big)$.
Let $T\subseteq Y$. If $f$ is surjective, then $f\big(f^{-1}(T)\big)=T$.