Tight measure

Let be a topological space, the Borel -field generated by the open sets and the paving (i.e. family of subsets) of all compact sets. A measure on is tight if

A finite tight measure on is a Radon measure. If is a separable complete metric space, every probability measure on is tight (Ulam's tightness theorem), [a2]. The terminology "tight" was introduced by L. LeCam, [a5].

More generally, let be two pavings on a set , and a set function defined on . Then is tight with respect to if

References

[a1]	P. Billingsley, "Convergence of probability measures" , Wiley (1968) pp. 9ff
[a2]	F. Topsøe, "Topology and measure" , Springer (1970) pp. xii
[a3]	K. Bichteler, "Integration theory (with special attention to vector measures)" , Lect. notes in math. , 315 , Springer (1973) pp. §24
[a4]	J.C. Oxtoby, S. Ulam, "On the existence of a measure invariant under a transformation" Ann. of Math. , 40 (1939) pp. 560–566
[a5]	L. LeCam, "Convergence in distribution of probability processes" Univ. of Calif. Publ. Stat. , 2 : 11 (1957) pp. 207–236