Definition:Lebesgue Integral

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Definition

Let $\lambda^n$ be a Lebesgue measure on $\R^n$.

Let $f: \R^n \to \overline \R$ be a Lebesgue integrable function, where $\overline \R$ denote the set of extended real numbers.


Then the $\lambda^n$-integral of $f$:

$\ds \int f \rd \lambda^n$

is called the Lebesgue integral of $f$.


The definition proceeds in several steps.


Lebesgue Integral of a Simple Function

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

Let $E$ be a measurable subset of $X$.

Let $\phi: E \to \overline \R$ be a $\Sigma$-measurable simple function which vanishes outside a set of finite measure.

Then the Lebesgue integral of $\phi$ on $E$ is defined as:

$\ds \int_E \map \phi x \rd \mu = \sum_{i \mathop = 1}^n a_i \map \mu {A_i}$

where $\ds \map \phi x = \sum_{i \mathop = 1}^n a_i \chi_{A_i}$.

If the term $0 \cdot \infty$ appears in the sum, the term is defined to be zero.


Lebesgue Integral of a Nonnegative Measurable Function

Let $f: E \to \overline \R$ be a nonnegative $\Sigma$-measurable function on $E \in \Sigma$.

Then the Lebesgue integral of $f$ is defined as:

$\ds \int_E f \rd \mu = \sup \set {\int_E \phi \rd \mu: \phi \text { is a simple function on E and } \phi \le f}$


Lebesgue Integral of a Measurable Function

Let $f: E \to \overline \R$ be a nonnegative $\Sigma$-measurable function on $E \in \Sigma$.

Let us define:

$\map {f^+} x := \max \set {x, 0}$
$\map {f^-} x := -\min \set {x, 0}$

Then $f^+$ and $f^-$ are positive measurable functions.


Let both $\map {f^+} x$ and $\map {f^-} x$ have finite Lebesgue integrals on $E$.

Then $f$ is Lebesgue integrable on $E$ and we define:

$\ds \int_E f \rd \mu = \int_E f^+ \rd \mu - \int_E f^- \rd \mu$.


Notation

The Lebesgue integral is frequently abbreviated as $\ds \int_E \phi$ or just $\ds \int \phi$.

If $E$ is the closed interval $\closedint a b$, we frequently write:

$\ds \int_a^b f = \int_E f$


Historically, the notations $\ds \int \map f x \rd x$ and $\ds \int f \rd x$ are used for the Lebesgue integral in place of the formally correct $\ds \int f \rd \lambda^n$.


Also see

  • Results about Lebesgue integrals can be found here.


Source of Name

This entry was named for Henri Léon Lebesgue.


Historical Note

The concept of a Lebesgue integral was inspired by Bernhard Riemann's own definition of an integral in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe of $1854$, on the subject of Fourier series.


Sources

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