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Using the continuum hypothesis, Sierpi nski constructed a non- measurable function f such that fh : f(x + h)6= f(x h)g is countable for every x: Clearly, such a function is symmetrically approximately continuous everywhere. Here we to show that Sierpi nski's example cannot be constructed in ZFC. Moreover, we show it is consistent with ZFC that if a function is symmetrically approximately continuous almost everywhere, then it is measurable.

Symmetrically approximately continuous functions, consistent density theorems, and Fubini type inequalities