Lusin's Theorem: Let $f$ be a real-valued measurable function on $E$. Then for each $\epsilon > 0$, there is a continuous function $g$ on $\Bbb R$ and a closed set $F$ contained in $E$ for which $f =g$ on $F$ and $m(E - F)< \epsilon$. Some extensions of Lusin's Theorem: a) Prove the extension o...

Let $(f_n)$ be a sequence of measurable functions on $E$ that converges to the real-valued $f$ pointwise on $E$. Show that $E=\bigcup_{k=1}^{\infty}E_k$, where for each index $k, E_k$ is measurable, and $(f_n)$ converges uniformly to $f$ on each $E_k$ if $k>1$, and $m(E_1)=0.$ My solution: Let $...