Tight measure
From Encyclopedia of Mathematics
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
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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
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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 |
How to Cite This Entry:
Tight measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tight_measure&oldid=28960
Tight measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tight_measure&oldid=28960