Dimension of Generic Reals

This paper investigates the Hausdorff measure of certain sets of generics in computability theory. Let $\mathrm{\Gamma }$ be the Turing ideal in which we take the dense open sets. The set of $\mathrm{\Gamma }$-Cohen generics has measure positive if and only if the gauge function is not dominated by every element in $\mathrm{\Gamma }$, under some mild restrictions on the gauge function. The set of $\mathrm{\Gamma }$-Mathias generics and the set of $\mathrm{\Gamma }$-Sacks generics have measure positive if and only if the gauge function eventually dominates every element in $\mathrm{\Gamma }$. This gives some comparison between the behavior of reals in the set and the measure of the set.