The Sobolev space with respect to different norms

Consider two different norms on the Sobolev space $W^{1,2}([0,1])$: the $L_2$ norm $$\|f\|_{L_2}^2:=\int_{[0,1]} f^2(x)\, dx$$ and the usual standard norm:

$$\|f\|_{H^1}^2:=\|f\|_{L_2}^2+\|f'\|_{L_2}^2.$$

It is obvious that this two norms are not equivalent and generate different topologies. Is it true that these two norms, nevertheless, generate the same Borel $\sigma$--algebra?

• How do norms generate a sigma algebra? – daw May 23 '18 at 20:32
• @daw thanks for your comment. I missed the word Borel $\sigma$--algebra. – Oleg May 23 '18 at 20:56

Since the $\sigma$-algebra generated by $\|\bullet\|_{H^1}$ is clearly finer than the one generated by the $L^2$-norm, it suffices that $\|\bullet\|_{H^1}$ is measurable with respect to the $L^2$-Borel-$\sigma$-algebra. For this, it suffices to show that $f \mapsto \|f'\|_{L^2}$ is measurable.
To show this, let $\Gamma \subset C_c^\infty ((0,1)) \setminus \{0\}$ be countable and such that $\Gamma \subset L^2$ is dense. Since $L^2$ is separable, and since $C_c^\infty$ is dense in $L^2$, such a set exists. It is not hard to see for any $h \in L^2$ that $$\|h\|_{L^2} = \sup_{g \in \Gamma} \frac{|\langle h, g \rangle|}{\|g\|_{L^2}} \, .$$
If we apply this to $f' \in L^2$ for $f \in H^1$, we see $$\|f'\|_{L^2} = \sup_{g \in \Gamma} \theta_g (f) \, ,$$ with $$\theta_g (f) := \frac{|\langle f', g \rangle|}{\|g\|_{L^2}} = \frac{|\langle f, g' \rangle|}{\|g\|_{L^2}} \, .$$ This identity show that $H^1 \to \Bbb{R}, f \mapsto \theta_g (f)$ is continuous (and hence measurable) with respect to $\|\bullet\|_{L^2}$.
As a countable supremum of measurable functions, the function $H^1 \to \Bbb{R}, f \mapsto \|f'\|_{L^2}$ is thus measurable.