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I am duplicating an old problem from stackexchange:

Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure (This is true in Carlson's model which is obtained by adding $\omega_2$ random reals to a model of CH here). Must there exist a Sierpinski set, i.e., an uncountable set of reals all of whose uncountable subsets are Lebesgue non null?

This is true in Carlson's model (the set of random reals added constitutes an $\omega_2$-sized Sierpinski set) and this is also true if there is a total extension of Lebesgue measure (via Gitik-Shelah theorem).

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