Consider a finite collection of sets. Every point in the union of all these sets is in at least two of these sets. Show that the measure of union of the sets is less than the sum of measures of each set, divided by two.
I understand the question and intiutionally I can think about what is happening. Since every point is in at least two of these sets, we are measuring over each point twice. However, I am having trouble providing a formal solution. Can anyone guide me to steps to solve the problem? Thanks in advance.
Say "the sets" are $E_1,\dots, E_n$. Let $E=\bigcup_{j=1}^n E_j$ and let $f_j=\chi_{E_j}$, the indicator function of $E_j$. The hypothesis says that $\sum f_j(x)\ge2$ for every $x\in E$, so $$\sum\mu(E_j)=\int_E\sum f_j\ge\int_E2=2\mu(E).$$