integrable majorant

An integrable majorant is the single umbrella that makes the dominated convergence theorem work. You have a whole sequence of functions wobbling about, and you want their limit and integral to cooperate. The trick is to find one fixed integrable function that lies above all of them in absolute value at once — a ceiling none of them ever pokes through. With such a ceiling, no area can sneak off to infinity.

Precisely, given measurable functions f_n, an integrable majorant is a function g with the integral of g dμ finite such that |f_n(x)| less than or equal to g(x) for all n and (almost every) x. The uniformity in n is the whole point: the same g must dominate the entire family simultaneously, not a different bound for each f_n.

Finding a majorant is the standard recipe for justifying any limit-integral interchange, including differentiation under the integral sign and termwise integration of a series (take g to be the sum of the absolute values). When no integrable majorant exists, the interchange may genuinely fail — that is exactly the situation in the moving-bump and tall-spike counterexamples, where the only natural bound is non-integrable.