Definition:Lebesgue Integral

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

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

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

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

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

or alternately

$$\int_{E} f \mathrm d\mu = \inf \{\int_{E} \phi \mathrm d\mu : \phi \text{ is a simple function on E and } \phi\geq f\}$$.

(It is a theorem that the two definitions coincide for measurable functions.)

The Lebesgue Integral of a Measurable Function
If $$f: E \to \R$$ 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 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 $$\int_{E} \phi\ $$ or just $$\int \phi$$. If $$E \ $$ is the closed interval $$[a,b] \ $$, we frequently write

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