The theory of Muckenhoupt's weight functions arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing $A_p$ weights from a measurably varying family of dyadic $A_p$ weights. This averaging process is suggested by the relationship between the $A_p$ weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Hölder ($RH_p$) conditions from families of dyadic $RH_p$ weights, and extends to the polydisc as well.
"Geometric-arithmetic averaging of dyadic weights." Rev. Mat. Iberoamericana 27 (3) 953 - 976, September, 2011.