Public Errata and Measurable Versions
These are public measurability and version conventions. General null-set and DCT proofs are in the integral companion; internal engineering records are excluded.
E. “Almost-Everywhere Limits” and Measurability
The main-text statements of monotone and dominated convergence include the assumption that the limiting function f is measurable. On an incomplete measure space, arbitrarily modifying a function on a subset of a null set need not leave it measurable. Merely being “equal almost everywhere to a measurable function” does not justify an unconditional assertion of measurability.
If we are given only measurable f_n converging outside a measurable null set, we can define the limit to be zero on the exceptional set and obtain a measurable version; for example, use the measurable upper limit and modify it on the measurable exceptional set. Integrals in nonnegative monotone convergence may take extended values, while an integrable dominating function in dominated convergence ensures that the limit is finite almost everywhere. For full proofs of nonnegative linearity, null-set modifications and DCT without σ-finiteness, see the integral foundations.
For nonnegative simple functions we consistently adopt 0\cdot\infty=0, so that a zero coefficient on a set of infinite measure does not create an undefined expression. For signed integrals, the positive- and negative-part integrals still cannot both be infinite.