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 $f : X \to [0, \infty]$ be a measurable function on a measurable space $(X, M, μ)$ .

Step 1: Simple Functions
A simple function $ϕ = \sum_{i = 1}^{n} a_{i} χ_{A_{i}}$ , where $A_{i} \in M$ , approximates $f$ from below. It is a linear combination of indicator functions
Step 2: Integral of Simple Function $\int_{X} ϕ d μ := \sum_{i = 1}^{n} a_{i} μ (A_{i})$
Step 3: Lebesgue Integral of $f$

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

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 $f (x) = χ_{Q \cap [0, 1]} (x)$ .

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