The Lebesgue Integral — Mathematics Knowledge System

Overview

The Lebesgue integral extends Riemann integration to a vastly larger class of functions, including all pointwise limits of integrable functions. Instead of partitioning the domain (as Riemann does), it partitions the range: the integral of f is built up by asking 'how much measure does f assign to values near y?' This reorientation, due to Henri Lebesgue (1902), resolved deep deficiencies of Riemann integration and became the foundation of modern analysis and probability.

Intuition

Slice the range of f into thin horizontal strips. For each strip at height y, measure the set of x-values where f(x) is approximately y, and multiply by the height of the strip. Summing these products over all strips gives the integral. For non-negative functions this is always well-defined (possibly infinite), and for functions with both positive and negative parts it is defined when the positive and negative parts are separately integrable.

Formal Definition

Definition

The Lebesgue integral is defined in three stages: for non-negative simple functions, for non-negative measurable functions, and for general measurable functions.

\int_X \varphi \, d\mu = \sum_{i=1}^n a_i \,\mu(E_i),\quad \varphi = \sum_{i=1}^n a_i \mathbf{1}_{E_i} \ge 0 \text{ simple}

Stage 1 — integral of a non-negative simple function

\int_X f \, d\mu = \sup\!\left\{\int_X \varphi \, d\mu : 0 \le \varphi \le f,; \varphi \text{ simple}\right\},\quad f \ge 0 \text{ measurable}

Stage 2 — integral of a non-negative measurable function

\int_X f\, d\mu = \int_X f^+\, d\mu - \int_X f^-\, d\mu

Stage 3 — integral of a general measurable function (when not of the form infty - infty)

f^+ = \max(f,0),\quad f^- = \max(-f,0),\quad f = f^+ - f^-,\quad |f| = f^+ + f^-

Positive and negative parts

f \in L^1(\mu) \iff \int_X |f|\, d\mu < \infty

f is integrable (in L^1) iff |f| has finite integral

Notation

Notation	Meaning
$\int_X f\, d\mu$	Lebesgue integral of f over X w.r.t. mu
$\int_A f\, d\mu$	Integral over a measurable subset A
$\int f\, d\lambda$	Lebesgue integral w.r.t. Lebesgue measure on R
$L^1(\mu)$	Space of mu-integrable functions

Properties

Linearity

\int_X (\alpha f + \beta g)\, d\mu = \alpha\int_X f\, d\mu + \beta\int_X g\, d\mu,\quad \alpha, \beta \in \mathbb{R}

Monotonicity

f \le g \text{ a.e.} \implies \int_X f\, d\mu \le \int_X g\, d\mu

Null sets do not matter

f = g \text{ a.e.} \implies \int_X f\, d\mu = \int_X g\, d\mu

Absolute bound

\left|\int_X f\, d\mu\right| \le \int_X |f|\, d\mu

Agreement with Riemann integral

\text{If } f \text{ is Riemann integrable on } [a,b], \text{ then the Lebesgue and Riemann integrals coincide.}

Worked Examples

The indicator 1_{[0,2]} is a simple function equal to 1 on E = [0,2] and 0 elsewhere.
$\mathbf{1}_{[0,2]} = 1 \cdot \mathbf{1}_{[0,2]} + 0 \cdot \mathbf{1}_{[0,2]^c}$
By the Stage 1 formula, the integral equals the coefficient times the measure of the set.
$\int_{\mathbb{R}} \mathbf{1}_{[0,2]}\, d\lambda = 1 \cdot \lambda([0,2]) + 0 \cdot \lambda([0,2]^c) = 1 \cdot 2 = 2$

Answer: The integral equals 2.

Practice Problems

Difficulty 6/10

Let mu be the counting measure on (N, 2^N). Show that ∫_N f dmu = sum_{n=1}^∞ f(n) for any f : N → [0,∞].

Difficulty 7/10

Prove: if f >= 0 is measurable and ∫ f dmu = 0, then f = 0 almost everywhere.

Difficulty 8/10

Prove linearity of the Lebesgue integral: ∫(f + g) dmu = ∫ f dmu + ∫ g dmu for non-negative measurable f, g.

Common Mistakes

Common Mistake

The Lebesgue integral just generalises Riemann with the same answers

While they agree on Riemann-integrable functions, the Lebesgue integral handles many more. For example, 1_Q has Lebesgue integral 0 but is not Riemann integrable. Convergence theorems (DCT, MCT) have no Riemann counterpart.

Common Mistake

∫ lim f_n = lim ∫ f_n always holds for Lebesgue integrals

Interchanging limit and integral requires additional conditions (DCT, MCT, or uniform integrability). The example f_n = n · 1_{(0,1/n)} shows the interchange can fail without a dominating function.

Common Mistake

If ∫ f dmu = 0 and f >= 0, then f = 0 everywhere

Only f = 0 almost everywhere is guaranteed. f can be nonzero on a null set.

Quiz

Historical Background

Lebesgue's key insight was that summing f(x_i)·Delta x_i (Riemann) is like paying a random set of coins one by one, while integrating by value (Lebesgue) is like sorting coins by denomination first. His 1902 thesis and 1904 book developed the theory. The integral enables the dominated convergence theorem and L^p space theory, essential to functional analysis, PDEs, and probability.

1902
Lebesgue's thesis introduces the Lebesgue integral
Henri Lebesgue
1904
Lecons sur l'integration published
Henri Lebesgue
1908
Fatou's lemma appears in Fatou's thesis
Pierre Fatou
1930s
Riesz representation theorem relates functionals to measures
Frigyes Riesz

Summary

The Lebesgue integral is built in stages: simple functions, then non-negative functions (via sup), then general functions via f = f^+ - f^-.
A function is in L^1(mu) iff ∫|f| dmu < ∞.
Key properties: linearity, monotonicity, null sets are irrelevant, |∫f| <= ∫|f|.
On Riemann-integrable functions the two integrals agree; Lebesgue handles vastly more functions.
Limit-integral interchange requires conditions: DCT (dominated convergence) or MCT (monotone convergence).

References

BookFolland — Real Analysis, 2nd ed. (1999), §2.1–§2.2
BookRudin — Real and Complex Analysis, 3rd ed. (1987), Chapter 1
WebsiteWikipedia — Lebesgue integration