Measured creatures

We prove that two basic questions on outer measure are undecidable. First we show that consistently every sup-measurable function f: ℝ² → ℝ is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functions f: ℝ² → ℝ the Cauchy problem y′ = f(x, y), y(x ₀) = y₀ has a unique almost-everywhere solution in the class AC₁(ℝ) of locally absolutely continuous functions on ℝ. Next we prove that consistently every function f: ℝ → ℝ is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous functions (from the reals to the reals) is dense in the space of arbitrary such functions. For the proofs we discover and investigate a new family of nicely definable forcing notions (so indirectly we deal with nice ideals of subsets of the reals - the two classical ones being the ideal of null sets and the ideal of meagre ones). Concerning the method, i.e., the development of a family of forcing notions, the point is that whereas there are many such objects close to the Cohen forcing (corresponding to the ideal of meagre sets), little has been known on the existence of relatives of the random real forcing (corresponding to the ideal of null sets), and we look exactly at such forcing notions.