2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/121570In order to develop a differential calculus for error propagation we study local Dirichlet forms on probability spaces with square field operator $Γ$ -- i.e. error structures -- and we are looking for an object related to $Γ$ which is linear and with a good behaviour by images. For this we introduce a new notion called the measure valued gradient which is a randomized square root of $Γ$. The exposition begins with inspecting some natural notions candidate to solve the problem before proposing the measure-valued gradient and proving its satisfactory properties.Probability31C25 65G99 60H07text