Exist $g:\mathbb{R}\to \mathbb{R}$ Lebesgue measurable and $h:[0, 1]\to \mathbb{R}$ Borel measurable such that $f = g \circ{}h$.

Question

For all $f:[0, 1]\to \mathbb{R}$ exist $g:\mathbb{R}\to \mathbb{R}$ Lebesgue measurable and $h:[0, 1]\to \mathbb{R}$ Borel measurable such that $f = g \circ{}h$.

Any ideas. Thanks

Answer 1

HINT: Let $h:[0, 1] \to[0, 1] $ given by $h(0) = 0$ and $ h (\sum_{n=1} ^{\infty}\frac{x_n}{2^n})=\sum_{n=1} ^{\infty} \frac{2x_n}{3^n} $ where the binary expansion has no "tail zeros". $h$ is strictly increasing and $ h([0, 1])$ is null. Let $g : [0, 1] \to \mathbb{ R} $given by : $g(x) = f(h^{-1}(x)) $ if $x \in h([0, 1])$ and $g(x)=0$ otherwise.