Construct an isometric isomorphism between $L_p(\mathbb{R})$ and $L_p[0;1]$

Question

I know that $L_p(\mathbb{R})$ and $L_p[0;1], \; p<+\infty$ are isometrically isomorphic, which means that there is an isomorphism that respects norm.

The question is how to construct it?

Answer 1

Consider a bijective function $\varphi\colon (0,1)\to\mathbb R$ with continuous derivative, and such that $\lim_{x\to 0}\varphi(x)=-\infty$ and $\lim_{x\to 1}\varphi(x)=+\infty$. Then define a map $$\Psi\colon \mathbb L^p(\mathbb R)\to \mathbb L^p((0,1)), \Psi\colon\left(x\mapsto f(x)\right)\mapsto \left(x\mapsto f\left(\psi(x)\right)\varphi'(x)\right).$$ Then we can check that $\Psi$ has the wanted properties.