To specify a measure on a Sigma-algebra it suffices to specify it on an Algebra (algebraic structure).
The measure then extends to the sigma algebra generated by that algebra, i.e. to the smallest sigma-algebra containing that algebra. The generation can be done by starting with the algebra and taking intersections and unions, so that it satisfies the axioms of a sigma-algebra. The generated sigma-algebra is unique, if the underlying measure is sigma-finite..