If $A, B \in [m,n] \subseteq \mathbb{R}$ are sets of measure zero, then is $AB = \{ab | a \in A, b \in B\}$ measure zero?

Question

If $A, B \in [m,n]$, where $[m,n] \subseteq \mathbb{R}$ are sets of measure zero, then is $AB = \{ab \; | \; a \in A, b \in B\}$ measure zero?

Answer 1

If $C$ is the Cantor set and $A=B=\{e^{x}:x\in C\}$ then $A$ and $B$ have measure $0$ but $AB$ contains all numbers of the form $e^{x+y}$ with $x,y \in C$. Since $C+C=[0,2]$ it follows that $AB$ does not have measure $0$. [I have used the following facts: $e^{x}$ is absolutely continuous on $[0,1]$ and any absolutely continuous function maps sets of measure $0$ to sets of measure $0$. Hence $A$ and $B$ have measure $0$]. Note also that $AB$ contains the entire interval $[e^{0},e^{2}]$ so it has positive measure.