Thierry Coquand and Bas Spitters have several papers on this topic, see <A HREF="http://arxiv.org/abs/0808.1522">Integrals and Valuations</A>, and <A HREF="https://pdfs.semanticscholar.org/1312/cf106c388a4138b59ed5f3cbe34d4c5241b2.pdf">Metric Boolean Algebras and Constructive Measure Theory</A>.