Let $\mathcal{M}$ be a sigma algebra on $X$. Let $f:X\to Y$. Define $ \mathcal{A}=\{B\subset Y : f^{-1}(B)\in \mathcal{M}\}. $ The problem is to show that $\mathcal{A}$ is a sigma algebra on $Y$.
This is my attempt.
Clearly, $\emptyset \in \mathcal{A}$.
Let $\{B_k\}_{k=1}^\infty$ be a countable collection of sets such that $f^{-1}({B_k})\in \mathcal{M}$. Then $f^{-1}(\cup B_k)=\cup f^{-1}(B_k)\in \mathcal{M}$. So $\cup B_k \subset Y$ and hence $\cup B_k\in\mathcal{A}$.
Also, $f^{-1}(B^c)=(f^{-1}(B))^c\in \mathcal{M}$ and so $B^c\in \mathcal{A}$.
Thus $\mathcal{A}$ is a sigma algebra on $Y$.
Please, is what I have done right?