Interpolation and inner maps that preserve measure

AbstractIt is shown, for n ⩾ m ⩾ 1, that there exist inner maps Φ: Bn → Bm with boundary values Φ∗: Bn → Bm such that σm(A) = σn(Φ∗−1(A)). where σn and σm are the Haar measures on ∂Bn and ∂Bm, respectively, and A ⊂ Bn is an arbitrary Borel set