Intersection of an uncountable number of almost sure events is almost sure?

Question

I know that an intersection of a countable number of almost sure events is almost sure. We can show this using De Morgan's laws. My question is whether an intersection of an uncountable number of almost sure events is also almost sure?

Answer 1

The intersection $\bigcap_{x\in [0,1]}([0,1]\setminus\{x\})=\emptyset$. The sets $[0,1]\setminus\{x\}$ have measure $1$ for the Lebesgue measure (probability). But $\emptyset$ has length zero.