Definition:Lebesgue Integral

The definition proceeds in two steps.

The Lebesgue Integral of a Simple Function
If $$\phi:\mathbb{R}\to \mathbb{R}$$ is a simple function which vanishes outside a set of finite measure, the Lebesgue Integral of $$\phi \ $$ is defined as

$$\int \phi(x)dx = \sum_{i=1}^n a_i m(A_i)$$

where $$m \ $$ is the Lebesgue measure and $$\phi(x) = \sum_{i=1}^n a_i \chi_{A_i}$$.

The integral is frequently abbreviated $$\int \phi$$.

The Lebesgue Integral of a Measurable Function
For any measurable function $$f:\mathbb{R} \to \mathbb{R}$$ defined on a set $$E \ $$ of finite measure, the Lebesgue Integral of $$f \ $$ is defined

$$\int_E f = \text{ inf} \int_E \psi \ $$

for all simple functions $$\psi \geq f \ $$. If $$E \ $$ is the closed interval $$[a,b] \ $$, we frequently write

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