I'm having some trouble with the following exercise:
Prove that the Cantor set $C$ has Lebesgue measure of zero.
This is what I did:
I will use $mA$ to denote the lebesgue measure of $A$.
We can express the Cantor set $C$ as: $$C=\bigcap _{n=1}^{\infty} C_n$$ where each $C_n$ is defined as the following union:
From the construction of the Cantor set, it's clear that $C_{n+1}\subseteq C_n$ and $mC_1=1$, so according to a proposition that we learned in class, $$mC=m \bigcap _{n=1}^{\infty} C_n=\lim_{n\to\infty} mC_n$$
So we have that $$mC_n=\sum_{k=0}^{3^{n-1}-1}\frac{2}{3^n}=\frac{2}{3^n}3^{n-1}=2/3$$
so $mC=\lim_{n\to\infty} mC_n=2/3$. This doesn't seam correct, but I'm not able to find the error in my proof. Where is this argument wrong?