Category:Definitions/Integrals of Integrable Functions

This category contains definitions related to Integrals of Integrable Functions.
Related results can be found in Category:Integrals of Integrable Functions.

Real Function

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

Let $f: X \to \overline \R$, $f \in \map {\LL^1} \mu$ be a $\mu$-integrable function.

Then the $\mu$-integral of $f$ is defined by:

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

where $f^+$ and $f^-$ are the positive and negative parts of $f$, respectively.

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.

Let $\map \Re f : X \to \R$ and $\map \Im f : X \to \R$ be the real part and imaginary part of $f$ respectively.

We define the integral of $f$ by:

$\ds \int f \rd \mu = \int \map \Re f \rd \mu + i \int \map \Im f \rd \mu$