Example where image of measurable set under measurable function is not measurable

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Let $(E,\mathscr{E})$ and $(F,\mathscr{F})$ be measurable spaces. Let $f:E\rightarrow F$ be a measurable relative to $\mathscr{E}$ and $\mathscr{F}$. When is $f(A)\notin\mathscr{F}$, where $A\in\mathscr{E}$?