In this section, the μ functional is introduced. It is defined as the functional whose value is the infinite-time average of the test function. The μ functional is formally characterized as the limit of approximating functions similar to the characterization of the Dirac delta as the limit of delta-approximating functions (Zemanian 1987, Section 1.3).
Let μ be the functional that associates to a test function ϕ its infinite-time average value. That is,
provided that the limit exists (and, hence, is independent of t).
In the following, the μ functional is heuristically characterized through formal manipulations. Let it be
where rect(t) = 1 for |t| ≤ 1/2 and rect(t) = 0 otherwise. For any finite T one has
Thus, in the limit as T→ ∞, we (rigorously) have
and we can formally write
with rhs independent of t, where μ(t) is formally defined as (see also (Silverman ...