Factorizations of Lebesgue measure via convolutions

Abstract

Given a continuous, increasing function ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) \phi :[0,\infty ) \to [0,\infty ) with ϕ ( 0 ) = 0 \phi (0) = 0 , we define the Hausdorff ϕ \phi -measure of a bounded set E E in the unit interval I = [ 0 , 1 ] I = [0,1] as H ϕ ( E ) = lim δ → 0 H ϕ , δ ( E ) {H_\phi }(E) = {\lim _{\delta \to 0}}{H_\phi }{,_\delta }(E) where H ϕ , δ E = inf ∑ i = 1 ∞ ϕ ( t i ) {H_\phi }{,_\delta }E = \inf \sum olimits _{i = 1}^\infty {\phi ({t_i})} and the infimum is taken over all countable covers of E E by intervals U i {U_i} with t i = | U i | = {t_i} = \left | {{U_i}} \right | = length of U i > δ {U_i} > \delta . We show that given any such ϕ \phi , there exist closed, nowhere dense sets E 1 , E 2 ⊂ I {E_1},{E_2} \subset I with H ϕ ( E 1 ) = H ϕ ( E 2 ) = 0 {H_\phi }({E_1}) = {H_\phi }({E_2}) = 0 and E 1 + E 2 ≡ { a + b : a ∈ E 1 , b ∈ E 2 } = I {E_1} + {E_2} \equiv \left \{ {a + b:a \in {E_1},b \in {E_2}} \right \} = I . The sets E i ( i = 1 , 2 ) {E_i}(i = 1,2) are constructed as Cantor-type sets E i = ⋂ n = 1 ∞ E i , n {E_i} = \bigcap olimits _{n = 1}^\infty {{E_{i,n}}} where E i , n {E_{i,n}} is a finite union of disjoint closed intervals. In addition, we give a simple geometric proof that the natural probability measures μ i {\mu _i} supported on E i {E_i} which arise as weak limits of normalized Lebesgue measure on E i , n {E_{i,n}} have the property that the convolution μ 1 ∗ μ 2 {\mu _1}*{\mu _2} is Lebesgue measure on I I .