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A discrete distribution for a discrete random variable X is a function f(A) which equals the probability that X=A for a set of distinct possible values of A, and 0 otherwise.
A continuous distribution for a continuous random variable X is a function f(x) which equals the probability that X<x for any x.

A linear map T : D → C which is continuous in the sense that

fn \underset D → f \implies Tfn → Tf.

The set of all distributions is written D , and we often write

Tf = 〈T,f 〉.

If g is a continuous function on Rⁿ , we may define the distribution T_g by

〈Tg,f 〉 = ⌠ ⌡ Rn  g(t)f(t) dt.

Often notational laziness lets us write

〈g,f 〉 = 〈Tg,f 〉

instead of T_g .