Theorem 4.5.4CE0D5F4DF7B2

Proof. In any inner product space weak lower semicontinuity of the norm follows from $\| z\| ^2=\lim \langle z_ j,z\rangle \le \| z\| \liminf \| z_ j\| $. Apply this to $z_ j=\mu _ j-\mu _ k$ to prove the strong-Cauchy/weak-limit assertion. If $p_ j=G\mu _ j\le G\mu _ k=p_ k$, then reciprocity gives

Thus monotone potentials give a strong Cauchy sequence: in the increasing case the norms are bounded by the limit norm, and in the decreasing case by the first norm, with the inequality direction reversed for the lower bound. Dominated or monotone integration of $p_ j$ against every finite-energy measure proves weak convergence to the prescribed limit.

For density, compact restrictions of $\mu $ converge strongly: their tail energies tend to zero by dominated convergence for the integrable kernel $G\, d\mu \, d\mu $. On each compact support, Lusin's theorem and the support-continuity argument split a measure of finite energy into disjoint compact restrictions with finite continuous potentials; finiteness is needed only almost everywhere for the original measure. Finite partial sums again converge strongly. For one such compact continuous potential $p$, let $D_ j$ be a regular exhaustion containing its support and put $q_ j=p$ outside $D_ j$, $q_ j=H^{D_ j}_ p$ inside. Sweeping makes $q_ j=G\nu _ j$ a potential, with compactly supported $\nu _ j$, and $q_ j\downarrow 0$. The preceding monotone assertion gives $\nu _ j\to 0$ strongly. Then $\mu -\nu _ j$ has compact support, and its Green potential $p-q_ j$ is continuous and compactly supported. Differences prove signed density.

Energy-bounded positive measures have uniformly bounded masses on each compact $K$: choose nonnegative $q\in C_ c^\infty $ equal to one on $K$, and use its signed finite-energy representing measure $\lambda $ above: $\mu _ j(K)\le \int q\, d\mu _ j\le \| \lambda \| \| \mu _ j\| $. Weak energy convergence therefore implies convergence on smooth compact tests and then on every continuous compact test by uniform approximation in a fixed compact neighborhood. Conversely a vague limit $\mu $ has $I(\mu )\le \liminf I(\mu _ j)$: insert compact cutoffs in both kernel variables, truncate the kernel, pass to product weak convergence, and increase cutoffs and truncations. Thus $\mu \in \mathcal E^+$. Vague convergence gives convergence against the dense compact-potential tests; Cauchy–Schwarz and the uniform energy bound extend it to every energy test, proving weak convergence.

Finally a strong Cauchy sequence in the positive cone has bounded energy and locally bounded masses. Diagonal compact measure selection gives a vaguely convergent subsequence and positive Radon limit. The preceding paragraph makes it an energy weak limit, and the strong-Cauchy assertion makes the whole sequence converge strongly. No completeness of the signed space was assumed. $\square$