Cutting a cake for infinitely many guests

Fair division with unequal shares is an intensively studied recourse allocation problem. For $i\in [n]$, let $\mu_i$ be an atomless probability measure on the measurable space $(C,\mathcal{S})$ and let $t_i$ be positive numbers (entitlements) with $\sum_{i=1}^{n}t_i=1$. A fair division is a partition of $C$ into sets $S_i\in \mathcal{S}$ with $\mu_i(S_i)\geq t_i$ for every $i\in [n]$. We introduce new algorithms to solve the fair division problem with irrational entitlements. They are based on the classical Last diminisher technique and we believe that they are simpler than the known methods. Then we show that a fair division always exists even for infinitely many players.