Majorization of regular measures and weights with finite and positive critical exponent

For the sets M∗p(R), 1 p < ∞, of positive finite Borel measures μ on the real axis with the set of algebraic polynomials P dense in Lp(R, dμ), we establish a majorization principle of their “boundaries,” i.e. for every μ ∈M∗p(R) there exists ν ∈M∗p(R) \ q>pM∗q (R) such that dμ/dν 1. A corresponding principle holds for the sets W∗p(R), p >0, of non-negative upper semi-continuous on R functions (weights) w such that P is dense in the space C0wp: For every w ∈ W∗p(R) there exists ω ∈W∗p(R) \ q