Exchanging limits and integrals

Let Riemann-integrable functions converge uniformly on a fixed finite interval. Their uniform limit is also Riemann integrable, and the integral of the difference is bounded by interval length times its supremum magnitude. This yields convergence of the integrals without locating any maximum error or calculating antiderivatives. The bound is quantitative: a tolerance on the desired integral determines a stricter tolerance for the function approximation. The interval must be the same for all functions.

The factor of interval length explains why the same argument cannot be carried over unchanged to an infinite interval. A uniform error of one over n across an interval of length n can retain unit area. Even on a finite interval, pointwise convergence alone gives no common height bound shrinking to zero. Before exchanging symbols, ask whether the integral error is controlled in a norm that actually tends to zero, rather than merely assuming that pointwise agreement must accumulate harmlessly.

On the unit interval, give a function height n on the open interval from zero to one over n and zero elsewhere. Every fixed positive input eventually lies outside that support, and the value at zero is always zero. Thus the functions tend pointwise to zero everywhere, but their integrals remain one. Mass concentrates in a narrowing region rather than disappearing. Each function is Riemann integrable, so the failure concerns the interchange, not the existence of the individual integrals.

On the nonnegative real line, instead use height one over n on the interval from zero to n. These functions tend uniformly to zero because their supremum heights vanish, yet their integrals again remain one. Here the error spreads over a domain whose effective length grows without bound. Finite-domain uniform convergence prevents neither scenario automatically when the hypotheses are changed: the spike violates uniform convergence, while the plateau violates the finite-length assumption of the earlier estimate.

Lebesgue dominated convergence applies to measurable functions fₙ on one measure space when fₙ tends to f almost everywhere and one integrable function g bounds every |fₙ| almost everywhere. The envelope g may not depend on n. It rules out uncontrolled concentration and escaping tails in a way suited to the measure space, and it yields convergence in integrated absolute error. Almost everywhere refers to the chosen measure; changing that measure can change which exceptions are negligible. This theorem will receive its construction and proof in the later measure-theory module.

For example, xⁿ on the unit interval is bounded by one and tends to zero except at the single endpoint one. Its integrals therefore tend to zero even though convergence is not uniform on that interval. The quadratic lab supports the simpler finite-interval estimate: predict how both its value and its displayed accumulated area change when a shrinks at fixed x. It cannot certify a Lebesgue dominator or reveal arbitrary concentrated spikes; those require an analytic argument outside its small model.