Definition:Lebesgue Integral

Definition
The definition proceeds in several steps.

The Lebesgue Integral of a Simple Function
Let $(X, \Sigma, \mu)$ be a measure space and let $E$ be a measurable subset of $X$.

If $\phi: E \to \overline{\R}$ is a $\Sigma$-measurable simple function which vanishes outside a set of finite measure, the Lebesgue Integral of $\phi$ on $E$ is defined as:


 * $\displaystyle \int_{E} \phi(x)\mathrm d\mu = \sum_{i=1}^n a_i \mu (A_i)$

where $\displaystyle \phi(x) = \sum_{i=1}^n a_i \chi_{A_i}$.

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

The Lebesgue Integral of a Nonnegative Measurable Function
If $f: E \to \overline{\R}$ is a nonnegative $\Sigma$-measurable function on $E\in\Sigma$, then the Lebesgue integral of $f$ is defined as


 * $\displaystyle \int_{E} f \mathrm d\mu = \sup \left\{{\int_{E} \phi \mathrm d\mu : \phi \text{ is a simple function on E and } \phi\leq f}\right\}$.

The Lebesgue Integral of a Measurable Function
If $f: E \to \overline{\R}$ is a nonnegative $\Sigma$-measurable function on $E\in\Sigma$, then define $f^+(x) = \max\{x, 0\}$ and $f^-(x) = -\min\{x, 0\}$.

Then $f^+$ and $f^-$ are positive measurable functions, and if they both have finite Lebesgue integral on $E$, we say that $f$ is Lebesgue integrable on $E$ and define


 * $\displaystyle \int_E f\mathrm d\mu = \int_E f^+\mathrm d\mu - \int_E f^-\mathrm d\mu$.

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

If $E$ is the closed interval $[a,b]$, we frequently write:
 * $\displaystyle \int_a^b f = \int_E f$