Additive set function properties I am reading an introduction to measure theory, which starts by defining $\sigma$-rings then additive set functions and their properties which are given without proof. I was able to prove two of them which are very easy $\phi(\emptyset)=0$ and $\phi(A_1\cup \ldots\cup A_n)=\displaystyle\sum_{i=1}^nA_i$.
There are three more which I couldn't do which are :
1- $\phi(A_1\cup A_2)+\phi(A_1 \cap A_2)=\phi(A_1)+\phi(A_2)$ for any two sets $A_1 $ and $A_2$.
2- If $\phi(A) \ge 0$ for all $A$ and $A_1 \subset A_2$ then $\phi(A_1) \le \phi(A_2)$.
3- If $A_1 \subset A_2$ and $|\phi(A_1)|< + \infty$ then $\phi(A_2-A_1)=\phi(A_2)-\phi(A_1)$.
Where $\phi$ is an additive set function.
I need some hints on how to proceed.
 A: I was able to use the third proposition to prove the first.
Let $A \subset B$. We have $(A \setminus B) \cap B = \emptyset$ then since $\phi$ is additive we have $\phi(A \setminus B) + \phi(B)=\phi((A-B) \cup B)=\phi(A) $ . Then $\phi(A \setminus B)=\phi(A)-\phi(B)$, this proves the second one.
Now for the first : $A \cup B= A\setminus (A\cap B) \cup B \setminus (A\cap B) \cup(A \cap B)$. Then $\phi(A \cup B)=\phi(A\setminus (A\cap B) \cup B \setminus (A\cap B) )= \phi(A)-\phi(A\cap B)+\phi(B)- \phi(A\cap B) + \phi(A \cap B)$ 
Hence upon adding $\phi(A \cap B) $ to both sides we get $\phi(A \cup B) + \phi(A \cap B)= \phi(A) + \phi (B)$. Can anyone help with the second one?
Edit: Here's the second one. Since $\phi(A) > 0$ for all sets then $\phi(A\setminus B)>0$ hence by the second property $\phi(A)-\phi(B) > 0$ when $B \subset A$.
A: I'm reading the same book I guess :). This is my try:
The core point is that $\phi (A_1 \cup A_2) = \phi [A_1 \cup (A_2 - A_1)]$ implies $\phi (A_2 - A_1) = \phi (A_1 \cup A_2) - \phi (A_1)$, by the properties you stated.
3 follows immediately, since
$A_1 \subset A_2$ implies $\phi (A_1 \cup A_2) = \phi(A_2)$.
2 follows from
$
0 \leq \phi(A_2 - A_1)
=
\phi (A_1 \cup A_2) - \phi (A_1)
=
\phi (A_2) - \phi (A_1)
$.
1 is a little more involved:
$
\phi(A_1 \cup A_2) + \phi(A_1 \cap A_2)
=
\phi[(A_1 - A_2) \cup A_2] + \phi[A_1 - (A_1 - A_2)]
=
\phi(A_1 - A_2) + \phi (A_2) + \phi[A_1 \cup (A_1 - A_2)] - \phi(A_1 - A_2)
=
\phi (A_2) + \phi[A_1 \cup (A_1 - A_2)]
=
\phi (A_2) + \phi(A_1)
$.
