Tightness of a probability measure ($\forall \epsilon >0\exists K\text{ compact}:\mu (K)>1-\epsilon$)

A probability measure is tight if compact sets carry arbitrarily nearly all its mass. Every Borel probability measure on a Polish space is tight. The stronger family condition is uniform tightness.

For càdlàg processes on a finite horizon, compact containment together with uniformly small increments over pairs of bounded stopping times whose separation tends to zero is a sufficient criterion for tightness in the Skorokhod J1 topology. The increment condition is in probability, not merely at deterministic times. If in addition the largest jump tends to zero in probability, every limiting law is supported on continuous paths.

The same compact set must work simultaneously for every measure in the family. Prokhorov's theorem relates this condition to relative compactness for the weak topology of probability measures on a Polish space.

A tight sequence of probability measures on the real line has a subsequence whose cumulative distribution functions converge at every continuity point of a cumulative distribution function. Choose a diagonal subsequence converging at every rational $q$, write those limits as $f(q)$, and define $F(x)={\inf }_{q>x}f(q)$. Monotonicity gives right continuity, and tightness of probability measures gives limits zero and one at the two ends of the real line. Bracketing by rational continuity approximations proves convergence at the continuity points of $F$. The corresponding probability measure is the weak subsequential limit: approximate a bounded continuous test function by step functions on a compact interval, and use tightness to bound the tails.