# Filter of Lebesgue measure one sets

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?

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First the easy part, the Cantor set is of the same cardinality as $[0,1]$ and has Lebesgue measure zero.
Finally, suppose $E$ is any subset, and $A$ is measurable with measure $1$, such that $m^*(E\cap A)=0$ then $m^*(E)=m^*(E\setminus A)\leq m^*([0,1]\setminus A)=0$.