Suppose that the elements of a set $A \subset [0, 1]$ have a property $\alpha$ and the elements of another set $B \subset [0, 1]$ have a property $\beta$, and that both sets have the measure $1$.
Then, is it necessarily true that their intersection $A \cap B$ has the measure $1$ and consists of elements which have both properties $\alpha$ and $\beta$ ?
Assume that $A$ and $B$ are subsets of $I=[0,1]$ with measure $1$.
Then the complements of $A$ and $B$ ($I-A$ and $I-B$) each have measure $0$. Now $I-(A\cap B)=(I-A)\cup(I-B)$, so $\mu(I-(A\cap B))=\mu((I-A)\cup(I-B))\le\mu(I-A)+\mu(I-B)=0$.
Therefore $\mu(A\cap B)=1$.
[spaceisdarkgreen answered the other part of your question in a comment.]