Lebesgue integral

Motivation

Unlike the Riemann integral, the Lebesgue integral allows us to integrate a wider class of functions, especially those with many discontinuities or defined over more abstract measure spaces.

Riemann: sums over partitioned domain intervals.
Lebesgue: sums over partitioned ranges (function values).
This "measure-then-integrate" approach gives better convergence properties and is central to modern analysis.

Definition (non-negative measurable function)

Let $(X, M, μ)$ be a measure space, and let

f : X \to [0, \infty]

be a measurable function, where $[0, \infty]$ is equipped with the Borel σ-algebra $B ([0, \infty])$ . That is, $f$ is measurable with respect to $M$ and $B ([0, \infty])$ .

We define the Lebesgue integral of $f$ in three steps:

Step 1: Simple Functions
A simple function is a finite linear combination of indicator functions:

ϕ = \sum_{i = 1}^{n} a_{i} χ_{A_{i}}, where a_{i} \in [0, \infty), A_{i} \in M .

We say that $ϕ$ approximates $f$ from below if $0 \leq ϕ (x) \leq f (x)$ for all $x \in X$ .

Step 2: Integral of a Simple Function
The integral of $ϕ$ with respect to $μ$ is defined by:

\int_{X} ϕ d μ := \sum_{i = 1}^{n} a_{i} μ (A_{i}) .

Step 3: Lebesgue Integral of $f$
The integral of $f$ is defined as the supremum over all simple functions that approximate it from below:

\int_{X} f d μ := sup {\int_{X} ϕ d μ | ϕ simple, 0 \leq ϕ \leq f} .

Extension to General Functions

For real-valued measurable $f$ , define:

f^{+} = max (f, 0), f^{-} = max (- f, 0)

Then,

\int_{X} f d μ := \int_{X} f^{+} d μ - \int_{X} f^{-} d μ

provided at least one of the terms is finite.

Example

Let $X = R$ with $μ$ the Lebesgue measure on $(R, B (R))$ .
Let $f (x) = χ_{Q \cap [0, 1]} (x)$ .

Riemann Integral: undefined (discontinuous everywhere). When you want to choose the height of each little rectangle, you don't know if it is 1 or 0.
Lebesgue Integral:

\int_{0}^{1} f d x = 0

since $Q \cap [0, 1]$ has measure zero.

Related: Lebesgue spaces