Example $\phi $ is continuous function $f_n \to f$ in measure. with $\phi \circ f_n\not\to \phi \circ f$ in measure
I am trying to find above example but unable to construct.
Also I tried following theorem
$\phi $ is uniformly continuous function $f_n \to f$ in measure. with $\phi \circ f_n\to \phi \circ f$ in measure
Let $\phi $ is uniformly continuous
$\forall \epsilon >0$ $\exists \delta >0$ such that $|x-y|<\delta\implies |\phi (x)-\phi (y)|<\epsilon $
On $E^c$ $|f_n(x)-f(x)|\leq \delta $$\implies |\phi \circ f_n(x)-\phi \circ f(x)||<\epsilon $
SO it is converges in measure .
Is there is any mistake ?
Please Help me to construct example