Absolutely Integrable

A function $f$ is absolutely integrable on a domain $D$ if

$$ \int_D |f| < \infty $$

Equivalently, $f \in L^1(D)$.

One Dimension

$$ \int_{-\infty}^{\infty} |f(x)| \, dx < \infty $$

Two Dimensions

$$ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} |f(x,y)| \, dx \, dy < \infty $$

Relation to Conditional Integrability

A function may be conditionally integrable (the integral $\int f$ converges) without being absolutely integrable. For example:

$$ \int_1^\infty \frac{\sin x}{x} \, dx \quad \text{converges, but} \quad \int_1^\infty \left|\frac{\sin x}{x}\right| dx = \infty $$

Absolute integrability implies conditional integrability, but not conversely.

Significance

Absolute integrability is a sufficient condition for the existence of the Fourier transform.