A symmetric density property for measurable sets (Q582404)

This is continuation of a previous authors' paper (summarized above). It is proved: Theorem 1. Let W and B be measurable subsets of a real interval whose union has full measure. If for each x, the set \(\{h>0:\quad x-h\in W\quad and\quad x+h\in B\}\) has density zero at zero, then these sets all have measure zero. Theorem 2. If a measurable real function has a nonnegative lower approximate symmetric derivative then it is non-decreasing on the set of points for which it is approximately continuous.