Dominated convergence and uniform integrability

The dominated convergence theorem assumes measurable fₙ converge almost everywhere to f and that one nonnegative integrable g bounds every |fₙ| almost everywhere. It concludes that f is integrable and that the L1 norm of fₙ−f tends to zero; convergence of the signed integrals follows. A different integrable bound for each n does not meet the hypothesis. Nor does a constant bound on an infinite-measure domain automatically provide an integrable envelope. The quantifiers and the domain are substantive parts of the theorem.

A proof can be organized around Fatou’s lemma. Apply it to nonnegative combinations involving the common envelope and the functions, or to 2g−|fₙ−f| after observing |f|≤g almost everywhere. Fatou then forces the upper limit of the L1 errors to be at most zero. Since these errors are nonnegative, they converge to zero. The envelope blocks both concentration and movement into regions where its own integral tail is small, explaining why domination is stronger than a bound on individual integral totals.

In this lesson, uniform integrability is used on a finite measure space. A family of integrable functions is uniformly integrable when the supremum of the integrals of their absolute values over the regions where those values exceed K tends to zero as K grows. The supremum is taken before the limit: a single cutoff must work for every family member. This condition implies a uniform L1 bound on a finite space, but the converse fails. The unit-area spikes n·1(0,1/n) have bounded L1 norms while all their mass remains above any fixed cutoff for sufficiently large n.

A common integrable envelope implies uniform integrability because the corresponding tails are bounded by tails of that envelope. Uniform integrability can hold even when no convenient common envelope is available. One frequently used sufficient condition is a uniform Lp bound for some p>1: on |f|>K, the inequality |f|≤|f|ᵖ/Kᵖ⁻¹ makes the tail integrals uniformly small. This argument gives a quantitative cutoff and exposes precisely why a bound only at p=1 lacks the necessary decay factor.

On a finite measure space, the Vitali convergence theorem says that convergence in measure together with uniform integrability gives L1 convergence to the limiting measurable function, which is then integrable. Convergence in measure means that, for each positive tolerance, the measure of the set where the error exceeds that tolerance tends to zero. Almost-everywhere convergence implies convergence in measure on a finite space, so Vitali also covers many familiar pointwise examples. Uniform integrability supplies exactly the control missing from the concentrating spike sequence.

To understand the mechanism, split the error into moderate values and a large-value tail. The moderate part is small except on an exceptional set whose measure vanishes; finite total measure bounds the remaining small error. Uniform integrability handles the tail uniformly. On the whole line, the functions (1/n) times the indicator of [0,n] provide a warning: their value tails vanish for K>1, and their uniform heights tend to zero, so they converge in measure to zero, but their L1 norms stay one. The finite-measure assumption, or an appropriate additional tightness condition in a broader formulation, cannot be dropped.