The standard, middle-thirds Cantor set can be thought of as the set of all numbers on the interval $[0, 1]$ whose ternary expansions contain no 1s, that is, numbers of the form $$\sum_{n=1}^{\infty} a_n 3^{-n} \text{ where } (a_n) \in \{0, 2\}^\mathbb{N}$$ It is well known that this Cantor set has Lebesgue measure $0$.
I am trying to show that a very similar-looking set, the set of points of the form $$\sum_{n=1}^{\infty} a_n e^{-n} \text{ where } (a_n) \in \{-1, 1\}^\mathbb{N}$$ also has Lebesgue measure $0$. I initially tried to take a bijection between these sets and show this bijection is a homeomorphism, but then realized homeomorphisms don't necessarily preserve measure.
Is there any way to use properties of the Cantor set to show that the latter set has measure $0$?