Suppose that $\mu$ is a measure on the Borel $\sigma$-algebra on $[0,1]$ and for every $f$ that is real-valued and continuously differentiable we have

$$ \left|\int f'(x)~\mu(\text{d}x)\right| \leqslant \left(\int_0^1 f^2(x) ~\text{d}x\right) ^{1/2}. $$

(1) Show that $\mu$ is absolutely continuous with respect to Lebesgue measure on [0,1].

(2) If $g$ is the Radon-Nikodym derivative of $\mu$ with respect to Lebesgue measure, prove that there exists a constant $c>0$ such that $$ \left|g(x)-g(y)\right|\le c\left|x-y\right|^{1/2},~~~~~~x,y\in[0,1]. $$

Attempt:

The inequality given reminds me of Jensen's inequality but I've no idea how to use it on (1).

Suppose that $\mu$ is a measure on the Borel $\sigma$-algebra on $[0,1]$ and for every $f$ that is real-valued and continuously differentiable we have

$$ \left|\int f'(x)~\mu(\text{d}x)\right| \leqslant \left(\int_0^1 f^2(x) ~\text{d}x\right) ^{1/2}. $$

(1) Show that $\mu$ is absolutely continuous with respect to Lebesgue measure on [0,1].

(2) If $g$ is the Radon-Nikodym derivative of $\mu$ with respect to Lebesgue measure, prove that there exists a constant $c>0$ such that $$ \left|g(x)-g(y)\right|\le c\left|x-y\right|^{1/2},~~~~~~x,y\in[0,1]. $$

Attempt

The inequality given reminds me of Jensen's inequality but I've no idea how to use it on (1).