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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).

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    $\begingroup$ This seems completely trivial: since $\|\cdot\|_{\dot{H}^1}$ is only a seminorm and $\|1\|_{\dot{H}^1} = 0$, then it follows immediately from the definition that in order to have $\|\sigma\|_{\dot{H}^{-1}} < \infty$ we must have $\langle \sigma, 1 \rangle = 0$. If it's that simple, I wouldn't think any reference should be needed. $\endgroup$ – Nate Eldredge Aug 7 at 19:58
  • $\begingroup$ I agree it's not too hard, but a reference would be helpful for my readers. $\endgroup$ – Amir Sagiv Aug 8 at 13:44

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