  In important papers on cake-cutting -- one of the key areas in fair division
and resource allocation -- the measure-theoretical fundamentals are not fully
correctly given. It is not clear (i) which family of sets should be taken for
the pieces of cake, (ii) which set-functions should be used for evaluating the
pieces, and (iii) which is the relationship between various continuity
properties appearing in cake-cutting.
  We show that probably the best choice for the familiy of subsets of $[0,1]$
is the Borel $\sigma$-algebra and for the set-function any `sliceable' Borel
measure. At least in dimension one it does not make sense to work with only
finitely additive contents on finite unions of intervals. For the continuity
property we see two possibilities. The weaker is the traditional divisibility
property, which is equivalent to being atom-free. The stronger is simply
absolute continuity with respect to Lebesgue measure. We also consider the case
of a base set (cake or pie) more general than $[0,1]$.
