Positivity preserving forms have the Fatou property

If (un)n∈IN is a sequence in L2(E; m) converging m-almost everywhere to u, then Fatou’s lemma says that (u, u)L2 ≤ lim infn(un, un)L2 , where we set (u, u)L2 = ∞ if u 6∈ L2(E; m). The corresponding result, where a Dirichlet form replaces the inner product, was used by Silverstein [5; Lemma 1.7] and by Fukushima, Oshima, and Takeda [2; Theorem 1.5.2] to define extended Dirichlet space and study time changes for symmetric Markov processes. However, their proofs require that E is a locally compact, separable, metric space, and that m is a positive Radon measure with full support. The purpose of this note is to drop the restrictions on E and m. In Proposition 1, we prove the Fatou property for symmetric Dirichlet forms, and in Proposition 2 we generalize the result to positivity preserving forms. The Fatou property simply means that the function v 7→ E(v, v) is lower semi-continuous on L2(E; m) with respect to the topology of convergence in m-measure. By way of comparison, the lower semicontinuity of v 7→ E(v, v) with respect to L2 convergence is equivalent (see [4]) to the form E being closed. Not every closed form has the Fatou property, but this stronger result holds when the form is positivity preserving.