ISL 1970 Problem 23

Let $E$ be a finite set, $P_E$ the family of its subsets, and $f$ a mapping from $P_E$ to the set of non-negative reals, such that for any two disjoint subsets $A,B$ of $E$ , $f(A\cup B)=f(A)+f(B)$ . Prove that there exists a subset $F$ of $E$ such that if with each $A \subset E$ , we associate a subset $A'$ consisting of elements of $A$ that are not in $F$ , then $f(A)=f(A')$ and $f(A)$ is zero if and only if $A$ is a subset of $F$ .