We view subsets of the natural numbers as their characteristic functions, which are elements of the Cantor space $2^\mathbb{N}$. We take the uniform probability measure on the Cantor space. Under this view, what is the measure of the family of all productive sets (in the sense of computability theory)? Immune sets? Sets which are neither immune nor productive nor computably enumerable?

Isn't this revised question just the one Andrej Bauer suggested and I answered in the comments on your earlier continuedfraction version? In the uniform measure on subsets of $\mathbb N$ (equivalent, if you replace sets with their characteristic functions, to the product measure on $\{0,1\}^{\mathbb N}$ arising from the uniform measure on $\{0,1\}$), the collection of immune sets has measure 1, and therefore the other two collections in your question have measure 0. 

