Triangle Inequality for Integrals

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Theorem

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

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

Then:

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

Corollary

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

$\ds \int \size f \rd \mu = 0$

Then:

$\ds \int f \rd \mu = 0$

Real Number Line

On the real number line, the takes the following form:

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.

Then:

$\ds \size {\int_a^b \map f t \rd t} \le \int_a^b \size {\map f t} \rd t$

Complex Plane

In the complex plane, the takes the following form:

Let $\closedint a b$ be a closed real interval.

Let $f: \closedint a b \to \C$ be a continuous complex function.

Then:

$\ds \size {\int_a^b \map f t \rd t} \le \int_a^b \size {\map f t} \rd t$

where the first integral is a complex Riemann integral, and the second integral is a definite real integral.

Complex Function

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

Let $\struct {\C, \map \BB \C}$ be the complex numbers made into a measurable space with its Borel $\sigma$-algebra.

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

Then $\cmod f$ is $\mu$-integrable and:

$\ds \cmod {\int f \rd \mu} \le \int \cmod f \rd \mu$

Proof

We have:

$\ds \size {\int f \rd \mu}$

$=$

$\ds \size {\int f^+ \rd \mu - \int f^- \rd \mu}$

Definition of Integral of Measure-Integrable Function

$\ds $

$\le$

$\ds \size {\int f^+ \rd \mu} + \size {-\int f^- \rd \mu}$

Triangle Inequality for Real Numbers, since $f$ is $\mu$-integrable both integrals are certainly real

$\ds $

$=$

$\ds \int f^+ \rd \mu + \int f^- \rd \mu$

$\ds $

$=$

$\ds \int \paren {f^+ + f^-} \rd \mu$

Integral of Positive Measurable Function is Additive

$\ds $

$=$

$\ds \int \size f \rd \mu$

Sum of Positive and Negative Parts

$\blacksquare$

Also see

Sources

2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $10.4 \ \text{(v)}$
1966: Walter Rudin: Real and Complex Analysis: $1$: Integration of complex functions: Theorem $1.33$