Factorizations of Lebesgue Measure via Convolutions

Abstract

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