TITLE: $\sigma$-field and family of sets
QUESTION [0 upvotes]: Let $f:[0,1] \to [0,1]$ be the function $f(x)=-4x^{2} + 4x$ let $\mathcal M$ be any $\sigma$-field in $[0,1]$.
Is the family of sets
$\mathcal N= \{f(A): A \in \mathcal M \} $ a $\sigma$-field in $[0,1]$? Why?

REPLY [3 votes]: Hint: If you look at the graph of the function, you will see that one can find a nonempty interval $I\subseteq [0,1]$ such that $f(I)\subsetneq[0,1]$, but $f(I^C)=[0,1]$. It follows that you get a counterexample by taking $\mathcal{M}=\Big\{\emptyset,[0,1], I, I^C\Big\}$, since $[0,1]\backslash f(I)$ will not be the forward image of any element of $\mathcal{M}$.
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