Triangle Inequality for Integrals

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f: X \to \overline \R$ be a $\mu$-integrable function.

Then:


 * $\ds \size {\int_X f \rd \mu} \le \int_X \size f \rd \mu$

Real Number Line
On the real number line, the Triangle Inequality for Integrals takes the following form:

Complex Plane
In the complex plane, the Triangle Inequality for Integrals takes the following form:

Also see

 * Absolute Value of Definite Integral
 * Modulus of Complex Integral