domain and support are not the same thing.
domainis most commonly used to describe the set of values for which a function (map, transformation, etc.) is defined. For example, a function f(x) that is defined for real values $x \in R$ has domain $\mathbb R$, and is sometimes said to be "a function over the reals."
supportof a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.
Consider the normal distribution, it seems that the domain is equal to the support, is it reasonable to say this? In another word, if I ask some mathematicians if the domain of the normal distribution is equal to its support, what would they say? Yes? or, They are not comparable?