I am reading lecture notes I borrowed from friend. It defines the filter of subsets of $[0,1]$ with Lebesgue measure $1$. Then it defines sets $A$ to be called $F$-stationary iff $A \cap Y \neq \varnothing$ for all $Y$ in the filter. Then it writes the $F$-stationary sets are the sets of positive outer measure.
Why is it outer measure and not Lebesgue measure? And why is it true? Does it hold that all subsets of $[0,1]$ with cardinality equal the cardinality of $[0,1]$ must have positive (outer or Lebesgue) measure?