I recently stumbled upon Wikipedia's page on convexity (http://en.wikipedia.org/wiki/Convex_function#Properties) and there's reference to Sierpinski's theorem from which we can deduce that for Lebesgue measurable functions mid-point convexity implies convexity. For reference: mid-point convexity means $$f\left(\frac{1}{2} x + \frac{1}{2} y\right) \leq \frac{f(x) + f(y)}{2} \ldotp$$
Now, the question is: how pathological has to be an example of a mid-convex function that is not convex? The usual examples usually do something like 'put one value on rationals, different on non-rationals' but that won't work here, since such a function is a.e. equal to a convex, constant function.
I understand that since the function can't be measurable it has to be pretty weird, but how weird exactly in this case?
The only related question I've found is this one: Mid-point convexity does not imply convexity but it doesn't contain any examples, there's just a remark mentioning Sierpinski's result in the comments.
