Measure extension theorem(unique) Please give an example of two probability measures $\mu \not = \nu$ on $\cal{F} $= all subsets of {1, 2, 3, 4} that agree on a collection of sets C with $\sigma(C)=\cal{F}$ . thanks in advance.

Obviously, $C$ should _not_ be a $\pi$-system ( _do you know why?_ ). Let me provide some collection $C$ that works and let you find $\mu$ and $\
u$ (there are infinitely many such pairs): $$C=\\{\\{1,2\\},\\{1,3\\},\\{1,2,3,4\\}\\}.$$