A Selection Theory for Multiple-valued Functions in the Sense of Almgren

A Qk(R ) -valued function is essentially a rule assigning k unordered and non necessarily distinct elements of R to each element of its domain set A ⊆ R . For a Qk(R) valued function f we construct a decomposition into k branches that naturally inherit the regularity properties of f . Next we prove that a measurable Qk(R ) -valued function admits a decomposition into k measurable branches. An example of a Lipschitzian Q2(R ) -valued function that does not admit a continuous decomposition is also provided and we state a selection result about multiple-valued functions defined on intervals. We finally give a new proof of Rademacher’s theorem for multiple-valued functions. This proof is mainly based on the decomposition theory and it does not use Almgren’s bi-Lipschitzian correspondence between Qk(R ) and a cone Q ⊂ R (n)k .