a region, $x$, (called a cake) is to be "sliced" so that each of a panel of m judges assess that the division as fair. each judge has his or her own measure, $ui(s)$ of the value of each part, $s$ of the cake and $ui(x)=1$. there are two settings. in the first the cake is to be distributed to two people so that every judge believes that the portions given to each recipients $(u and x-u)$ is worth exactly $½$ (i.e. $ui(u)= ½ = ui(x-u) for all i=1,2, … , m)$. in the second setting, the m $(m > 2)$ judges are taking a portion of the cake (i.e., $ui$) for themselves. they want a division of the cake (i.e., $uj \cap uk$ is empty for each $j\neq k$ and $u1 +u2+ … +um = x$) so that each believes they received more than their fair share of the cake (i.e. $ui(ui) > 1/m for each i)$. both settings have solutions. the solutions give an introduction to measure theory and a fixed point theorem.