Lebesgue integral

Overview

A way to “fix” the Riemann integral, which is not able to integrate some functions. Instead of using partitions based on smaller closed intervals, it uses partitions of measurable sets.

Definition

For a measure space $(X,\mathcal{S},\mu )$, and a measurable function $f:X\to [0,\infty ]$ the the Lebesgue integral (or integral with respect to a measure) is

$$\int fd\mu =\sup \{\mathcal{L}(f,P)|\text{P is an}\mathcal{S}\text{-partition of X}\}$$

Where $\mathcal{L}$ is the lower Lebesgue sum of $f$ on $P$.

Notes about more general functions

Since the definition given only is defined for non-negative functions it would be pretty restrictive. However, we can split the function f into the negative and positive parts, thus for a general function

$$\int fd\mu =\int f^{+}d\mu -\int |f^{-}|d\mu$$

Also, the Lebesgue integral can be found using a process of approximation by simple functions.

Properties of Lebesgue integration

Order preserving

linear

Absolute value

$$\left|\int fd\mu \right|\le \int |f|d\mu$$

Links

• Lebesgue integrability
• integral leaks