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A problem related to the Continuum Hypothesis which was solved by Solovay (1970) using the Inaccessible Cardinals Axiom. It has been proven by Shelah and Woodin (1990) that use of this Axiom is essential to the proof.
See also Continuum Hypothesis, Inaccessible Cardinals Axiom, Lebesgue Measure
References
Shelah, S. and Woodin, H. ``Large Cardinals Imply that Every Reasonable Definable Set of Reals is
Lebesgue Measurable.'' Israel J. Math. 70, 381-394, 1990.
Solovay, R. M. ``A Model of Set-Theory in which Every Set of Reals is Lebesgue Measurable.'' Ann. Math. 92, 1-56, 1970.