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I study the subject of fair division (cake-cutting), and many papers contain a reference to a theorem by Lyapunov, which states that the range of any real-valued, non-atomic vector measure is compact and convex.

Can you recommend an online resource that can help me understand this theorem in an intuitive fashion, without having to read an entire book or take an entire course?

Note: I am not looking to become an expert in this field; I just want to get some intuition about this theorem, so that I can understand the papers that rely on it.

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The most famous and most elegant proof of Lyapunov's vector measure theorem is to be found in

Joram Lindenstrauss, A short proof of Liapounoff’s convexity theorem. J. Math. Mech., 15:971–972, 1966.

The proof requires, however, some knowledge of functional analysis and the paper is not that easy to get your hands on. An elementary proof of the result can be found in:

David A. Ross, An Elementary Proof of Lyapunov's Theorem. The American Mathematical Monthly , Vol. 112,7:651-653, 2005

That paper gives also references to some other proofs.

Most proofs of Lyapunov's theorem I know are highly nonconstructive. For a relatively constructive proof, see

Alan Hoffman, Uriel G. Rothblum, A proof of the convexity of the range of a nonatomic vector measure using linear inequalities, Linear Algebra and its Applications, Volume 199, Supplement 1, 1 March 1994, Pages 373-379

For an alternative proof of the one-dimensional case used in that paper, see here.

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