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.