Lebesgue measure of an intersection of four sets that are contained in [0,1] Let $A_1,A_2,A_3, A_4$ be measurable subsets of $[0,1]$, such that $\displaystyle\sum_{k=1}^{4}m(A_k)>3$. Prove that
$$
m\left(\bigcap_{k=1}^{4}A_k\right)>0.
$$

Let the superscript $c$ denote the complement in $[0,1]$.  Recall that $m(A)=1-m(A^c)$.  Then $$ m\left(\bigcap_{i=1}^4A_i\right)=1-m\left(\bigcup_{i=1}^4A_i^c\right)\geq 1-\sum_{i=1}^4 m(A_i^c)$$
$$=\sum_{i=1}^4 m(A_i)-3>0.$$
The last inequality follows from our starting asumption.
