Absolute Continuity of the Integral

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space.

(i) Nonnegative functions. Let $f\in\mathcal{M}^{+}$ with $\int_{X}f\,d\mu<\infty$. For every real $\varepsilon>0$ there is a real $\delta>0$ such that every $A\in\mathcal{F}$ with $\mu(A)<\delta$ satisfies

$$\int_{A}f\,d\mu<\varepsilon .$$

(ii) Integrable functions. Let $f\in\mathcal{L}^{1}(\mu)$ (then $|f|\in\mathcal{L}^{1}(\mu)$ by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (x), and the integrals over $A$ below are defined by Integrals over Measurable Subsets (vi)). For every real $\varepsilon>0$ there is a real $\delta>0$ such that every $A\in\mathcal{F}$ with $\mu(A)<\delta$ satisfies

$$\int_{A}|f|\,d\mu<\varepsilon\qquad\text{and}\qquad\Bigl|\int_{A}f\,d\mu\Bigr|<\varepsilon .$$