Cutting a Cake for Infinitely Many Guests

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