# Homogeneous negative Sobolev space is zero-mean

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 $$\|\sigma \|_{\dot{H}^{-1}}:\,=\sup\limits_{\|f\|_{\dot{H}^1}\leq 1} |<\sigma ,f>| < \infty \, \Rightarrow \sigma (\Omega )=0 \, ,$$ where $$\|f\|_{\dot{H}^1}:\,=\|\nabla f\|_2$$, 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?