tightness of a family of measures
On a non-compact space, mass can run away to infinity and a sequence of measures can fail to have any weak limit, even after passing to subsequences. Tightness is the condition that rules out this escape of mass. It is the single most important hypothesis in advanced limit theory, because it is precisely what upgrades 'these measures look like they should converge' into 'a convergent subsequence actually exists'.
A family Pi of probability measures on a metric space S is tight if for every epsilon > 0 there is a compact set K = K_epsilon in S with mu(K) >= 1 - epsilon for all mu in Pi. The single compact set must work uniformly over the whole family: no matter which measure you pick, at most epsilon of its mass lives outside K. On R, a family is tight iff for every epsilon there is a finite interval [-M, M] capturing at least 1 - epsilon of every measure's mass; the family fails to be tight exactly when mass can be pushed off to plus or minus infinity. On a Polish space, a SINGLE probability measure is automatically tight (this is Ulam's theorem), so tightness is genuinely a statement about the uniformity across a family.
Tightness matters because of Prokhorov's theorem: on a Polish space, a family is tight if and only if it is relatively compact in the weak topology, meaning every sequence from it has a weakly convergent subsequence. So tightness is the workhorse you verify to guarantee subsequential limits exist; you then identify the limit separately (often by computing characteristic functions or finite-dimensional distributions). This two-step pattern, prove tightness, then identify the limit, is the standard architecture of every functional limit theorem, including Donsker's invariance principle.
Suppose X_n are random variables with sup_n E[|X_n|] = C < infinity. By Markov's inequality, P(|X_n| > M) <= C/M for all n, so taking M = C/epsilon gives P(|X_n| in [-M, M]) >= 1 - epsilon uniformly in n. Hence the laws of X_n are tight: a uniform first-moment bound is enough to guarantee subsequential weak limits exist.
A uniform moment bound implies tightness via Markov; this is the most common way tightness is verified in practice.
Tightness guarantees a convergent SUBSEQUENCE, not convergence of the whole sequence. To conclude mu_n => mu you also need all subsequential limits to coincide, usually shown by matching characteristic functions or finite-dimensional laws.