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Map from power set to the same power set

Source: 26th annual VJIMC (2016), Category II, Problem 2

April 10, 2016
college contestsset theory

Problem Statement

Let XX be a set and let P(X)\mathcal{P}(X) be the set of all subsets of XX. Let μ:P(X)→P(X)\mu: \mathcal{P}(X) \to \mathcal{P}(X) be a map with the property that μ(A∪B)=μ(A)∪μ(B)\mu(A \cup B) = \mu(A) \cup \mu(B) whenever AA and BB are disjoint subsets of XX. Prove that there exists F⊂XF \subset X such that μ(F)=F\mu(F) = F.
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