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I have this funny three part question about the Cantor function and Lebesgue measure.

a) Suppose $g:\mathbb{R}\rightarrow\mathbb{R}$ be continuous and $h:\mathbb{R}\rightarrow \mathbb{R}$ be Borel measurable. Then $h\circ g$ is Borel measurable.

b) Find a continuous function $g:\mathbb{R}\rightarrow \mathbb{R}$ and a Lebesgue measurable function $h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h\circ g$ is not Lebesgue measurable.

c) Find two a.e. continuous functions $g,h:\mathbb{R}\rightarrow \mathbb{R}$ such that $g\circ h$ is discontinuous everywhere.

I can solve part a) easily, as for any set $B=(-\infty,b)$ we have $(h\circ g)^{-1}(B)=g^{-1}(h^{-1}(B))$ and so, using the given properties, $(h\circ g)^{-1}(B)$ is a Borel set.

Part b) is troublesome. I suspect $g$ is the Cantor function but I cannot pinpoint what $h$ is. Possibly h will involve a non Lebesgue measurable set... Same with part c), I am at a loss... Thanks in advance for any help!

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  • $\begingroup$ An Idea for b). Can I use characteristic functions? $\endgroup$ – MelaniesWoes Apr 7 '18 at 20:06

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