Let α > 0. The operator of the form

formula here

is considered, where the real weight function v(x) is locally integrable on R+ := (0, ∞). In case v(x) = 1 the operator coincides with the Riemann–Liouville fractional integral, Lp → Lq estimates of which with power weights are well known. This work gives Lp → Lq boundedness and compactness criteria for the operator Tα in the case 0 < p, q < ∞, p > max(1/α, 1).

It is shown, by a simple and direct proof, that if a bounded valuation on a monotone convergence space is the supremum of a directed family of simple valuations, then it has a unique extension to a Borel measure. In particular, this holds for any directed complete partial order with the Scott topology. It follows that every bounded and continuous valuation on a continuous directed complete partial order can be extended uniquely to a Borel measure. The last result also holds for σ-finite valuations, but fails for directed complete partial orders in general.