Bands of invariantly extensible measures

Given two σ-algebrasU ⊂A, invariant under a fixed semigroupG of transformations, the following subsetC of the lattice coneM (U) _G ofG-invariant finite measures onU is shown to be (the positive part of) a band inM (U) _G : AG-invariant measure μ belongs toC iff the setexM Bμ) _G of extremalG-invariant extensions of μ toB is non-empty and eachG-invariant extensionv of μ admits a barycentric decompositionv=→v′ρ(dv′) with some representing probability ρ onexM U μ) _G .—Any band of extensible measures allows to study the corresponding extension problem locally.