absolute continuity of the integral

An integrable function cannot hide much of its mass in a tiny place. Absolute continuity of the integral makes this quantitative: if you only integrate over a set of small enough measure, the contribution is guaranteed small — uniformly, no matter where that small set sits. Squeeze the domain down to nearly nothing and the integral over it goes to zero with it.

Precisely, if f is integrable then for every epsilon greater than 0 there is a delta greater than 0 such that for every measurable set E with μ(E) less than delta, the integral over E of |f| dμ is less than epsilon. The point is uniformity in E: one delta works for all small sets at once, controlled only by their measure. This is what justifies the phrase absolutely continuous (with respect to the measure).

This property fails for non-integrable functions and is the analytic engine behind several results: the indefinite integral x maps to the integral of f over [a, x] is an absolutely continuous function of x, and the set function E maps to the integral of f over E is an absolutely continuous measure. It is the integral-side shadow of the Radon–Nikodym relationship.