Another example is any of the denegenerate solutions of the Cauchy functional equation http://en.wikipedia.org/wiki/Cauchy's_functional_equation#Properties_of_other_solutions  , which has also the property of being measurable; unfortunately, I don't know when this condition is achieved, due to the not-so-easy construction of the function: maybe using the linearity one can show something, I don't know...

However this class of function suggests a solution to your problem, for example $f(a+ \sqrt{2} b ) = a - \sqrt{2} b$ whenever $a,b \in \mathbb{Q}$ and $f \equiv 0$ out of $\mathbb{Q} ( \sqrt{2} )$. In this way you have a measurable function (because it is $0$ out of a countable set) and it has the property of the dense graph.

Another example is any of the denegenerate solutions of the Cauchy functional equation (Wikipedia: Cauchy's functional equation - Properties of other solutions), which has also the property of being measurable; unfortunately, I don't know when this condition is achieved, due to the not-so-easy construction of the function: maybe using the linearity one can show something, I don't know...

However this class of function suggests a solution to your problem, for example $f(a+ \sqrt{2} b ) = a - \sqrt{2} b$ whenever $a,b \in \mathbb{Q}$ and $f \equiv 0$ out of $\mathbb{Q} ( \sqrt{2} )$. In this way you have a measurable function (because it is $0$ out of a countable set) and it has the property of the dense graph.