The following fact seems to be well-known and easy to prove: Let $\sigma$ be a Borel measure on a Borel set $\Omega \subseteq \mathbb{R}^d$, then $$\|\sigma \|_{\smash{\dot{H}}^{-1}}:\,=\sup\limits_{\|f\|_{\smash{\dot{H}}^1}\leq 1} |\langle\sigma ,f\rangle| < \infty \, \Rightarrow \sigma (\Omega )=0 \, ,$$ where $\|f\|_{\smash{\dot{H}}^1}:\,=\|\nabla f\|_2$, and the inner product is just the integral of the product.

However, I couldn't find a proper reference in standard Sobolev Spaces textbooks (I tried Adams & Fournier and Leoni). Is this wrong? Does anyone know a solid reference?

(cross posted from MSE after a week with no answer).