DOUBLING MEASURES AND NONQUASISYMMETRIC MAPS ON WHITNEY MODIFICATION SETS IN EUCLIDEAN SPACES

Let E be a closed set in R-n and W a Whitney decomposition of R-n\E. Choosing one point from the interior of each cube in W we obtain a set F and then we say that the set E boolean OR F is a Whitney modification of E. The Whitney modification of a measure mu on R-n to E boolean OR F is a measure nu defined on E boolean OR F by nu equivalent to mu on E and by nu({x}) = mu(I-x) for every x is an element of F, where I-x is an element of W is the cube containing the point x. We prove that a measure on E boolean OR F is doubling if and only if it is the Whitney modification of a doubling measure on R-n. As its application, we show that there are metric spaces X,Y and a nonquasisymmetric homeomorphism f of X onto Y such that a measure mu on X is doubling if and only if its image mu circle f(-1) is doubling on Y.