If $m(A)+m(B)>1$, then $A\cap B\neq \varnothing$

Question

A question from a test in the theory of measure from last year that I do not find a solution to:

Let $A,B\subset [0,1]$ be lebesgue measurable sets.

A. If $m(A)+m(B)>1$, then $A\cap B\neq \varnothing$.

B. Give an example of a measure space $(X,\mathcal{B},\mu)$ and measurable disjoint sets $A,B\in \mathcal{B}$ that satisfy $\mu (A)=\mu (B)=\mu (X)$.

I tried to start with $m([0,1])=1$ and I got stuck. Can someone help me?

Answer 1

Let $A,B$ be two subsets of $[0,1]$. Then $$m(A\cup B)=m(A)+m(B)-m(A\cap B)>1-m(A\cap B)$$

Since $A,B\subseteq[0,1]$, this means that $m(A \cup B)\leq 1$. From the first and the second results : $$m(A\cup B)-m(A\cap B)>1 \text{ and } m(A \cup B)\leq 1$$ we get that $m(A\cap B)\neq 0$ which means that $A\cap B\neq \emptyset$.