Construct a function $f : [0, 1] \to [0, 1]$ as follows. Let $\{I_n\}$ be an enumeration of the open intervals in [0, 1] having rational endpoints. For each $n \in \mathbb N$, let $K_n \subset I_n$ be a Cantor set of positive Lebesgue measure such that the sequence $\{K_n\}$ is pairwise disjoint and $\sum_{n=0}^\infty λ(K_n ) = 1$. Define $f_n$ on $K_n $ to be continuous on $K_n $, nondecreasing, and such that $f_n (K_n ) = [0, 1]$. Let
$f(n) = \begin{cases} f_n (x), & \text{if $$x$ \in k_n$} \\ 0, & \text{if $x$ $\in [0,1]$ \ $\cup_{n=0}^\infty k_n$} \end{cases}$
(a) Show that f is Lebesgue measurable.
(b) Show that f (I) = [0, 1] for every open interval $I \subset [0, 1]$.
(c) Using the sets $K_n $, find continuous functions on [0, 1] that approximate f in the Lusin sense.
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