Most of what I've read defines the support of a distribution topologically if it defines support precisely; e.g the smallest closed subset that has full measure (i.e. the intersection of all closed subsets of full measure). I've seen allusions to purely measure-theoretic definitions, and I think I even came across one once, but I can't find it. Is there a measure-theoretic definition of support that everyone agrees on?
Here is a comment made by Dmitri Pavlov for this MathOverFlow question. Maybe it will be helpful to you.