Let \(n\geq 1\) and \(h:(0,1)\rightarrow (0,\infty )\) be a function such that \( r^{n+1}h(r)\) is increasing and bounded, and \(\int_{0}^{1}h(r)\,dr=\infty \). A kernel \(H\) is defined on \(\mathbb{R}^{n}\) by \(H(x)=\int_{| x| }^{1}h(r)\,dr\) when \(\left| x\right| \leq 1/2\) and by \( H(x)=H(x_{0})\) when \(\left| x\right| >\left| x_{0}\right| =1/2\). The main result of this paper establishes separate necessary and sufficient conditions, in terms of the modulus of continuity \(\omega (\mu ,r) \) of a measure \(\mu \), for the energy integral \(\iint H(s-t)\,d\mu (t)\,d\mu (s)\) to be finite. The necessary condition is that \(\int_{0}^{1}h(r)\omega (\mu ,r)^{2}\,dr<\infty \), while the sufficient condition is that \( \int_{0}^{1}h(r)\omega (\mu ,r)\,dr<\infty \)\ The proofs involve studying the behaviour of the Poisson integral of \(\mu \) in the half-space \(\mathbb{R} ^{n}\times (0,\infty )\). Some of the results are shown to be sharp.