Convergence Theorems — Mathematics Knowledge System

Overview

The three classical convergence theorems — the Monotone Convergence Theorem (MCT), Fatou's Lemma, and the Dominated Convergence Theorem (DCT) — are the most powerful and frequently used results in integration theory. They provide conditions under which the limit of integrals equals the integral of the limit, resolving the fundamental question of when limit and integral can be interchanged.

Intuition

MCT: if functions increase monotonically to a limit, their integrals increase to the integral of the limit — no surprises possible when everything is going up. Fatou: even without monotonicity, the integral of the limit cannot exceed the liminf of the integrals — a one-sided inequality that costs nothing to assume. DCT: if all functions are dominated by a single integrable function (a referee), the integral and limit can be freely exchanged.

Formal Definition

Definition

All three theorems concern sequences of measurable functions on a measure space (X, F, mu).

\text{MCT: } 0 \le f_1 \le f_2 \le \cdots,; f_n \to f \text{ a.e.} \implies \int_X f_n\, d\mu \nearrow \int_X f\, d\mu

Monotone Convergence Theorem

\text{Fatou: } f_n \ge 0 \text{ a.e.} \implies \int_X \liminf_{n\to\infty} f_n\, d\mu \le \liminf_{n\to\infty} \int_X f_n\, d\mu

Fatou's Lemma

\text{DCT: } |f_n| \le g \text{ a.e., } g \in L^1(\mu),; f_n \to f \text{ a.e.} \implies \lim_{n\to\infty}\int_X f_n\, d\mu = \int_X f\, d\mu

Dominated Convergence Theorem

\text{DCT also gives: } \lim_{n\to\infty}\int_X |f_n - f|\, d\mu = 0

DCT — L^1 convergence

Notation

Notation	Meaning
$f_n \to f \text{ a.e.}$	Pointwise convergence almost everywhere
$\liminf_{n\to\infty} f_n$	Limit inferior of a sequence of functions
$\nearrow$	Monotone increasing convergence
$\\|f_n - f\\|_1 \to 0$	Convergence in mean (L^1 norm)

Worked Examples

Define f_n = e^{-x} · 1_{[0,n]}. Then f_n ↗ e^{-x} pointwise on [0,∞).
$f_n(x) = e^{-x}\mathbf{1}_{[0,n]}(x) \nearrow e^{-x}$
Each f_n is Riemann integrable, so ∫ f_n dlambda = ∫_0^n e^{-x} dx = 1 - e^{-n}.
$\int_{[0,\infty)} f_n\, d\lambda = \int_0^n e^{-x}\, dx = 1 - e^{-n}$
By MCT, ∫_{[0,∞)} e^{-x} dlambda = lim_{n→∞} (1 - e^{-n}) = 1.
$\int_{[0,\infty)} e^{-x}\, d\lambda = \lim_{n \to \infty}(1 - e^{-n}) = 1$

Answer: ∫_{[0,∞)} e^{-x} dlambda = 1.

Practice Problems

Difficulty 7/10

Use MCT to prove that if f >= 0 is measurable, then ∫ f dmu = lim_{n→∞} ∫ min(f, n) dmu.

Difficulty 7/10

Why does DCT fail for f_n = n · 1_{(0,1/n)} but succeed for g_n = (sin(x/n))/(x) · 1_{[1,∞)}?

Difficulty 8/10

Derive Fatou's Lemma from the Monotone Convergence Theorem.

Common Mistakes

Common Mistake

DCT requires uniform convergence

DCT only requires pointwise a.e. convergence together with an integrable dominating function. Uniform convergence would suffice but is far stronger than needed.

Common Mistake

Fatou's Lemma is an equality

Fatou gives only the inequality ∫ liminf f_n <= liminf ∫ f_n. Strict inequality is possible (e.g., f_n = 1_{[n,n+1]}).

Common Mistake

MCT works for decreasing sequences

MCT requires an increasing sequence. For decreasing sequences one needs the additional hypothesis that the first function is integrable (analogously to continuity from above for measures).

Quiz

Historical Background

The Monotone Convergence Theorem was essentially known to Lebesgue in his 1902 thesis. Fatou's lemma appeared in Pierre Fatou's 1906 thesis on trigonometric series. The Dominated Convergence Theorem, the most widely applied of the three, was proved by Lebesgue and became the theoretical cornerstone enabling modern Fourier analysis, PDE theory, and probability.

1902
Lebesgue proves the Monotone Convergence Theorem implicitly
Henri Lebesgue
1906
Fatou's lemma appears in Fatou's thesis on trigonometric series
Pierre Fatou
1910
Dominated Convergence Theorem formalised by Lebesgue
Henri Lebesgue

Summary

MCT: for 0 <= f_n ↗ f a.e., we have ∫ f_n dmu ↗ ∫ f dmu (no integrability assumption needed).
Fatou's Lemma: for f_n >= 0 a.e., ∫ liminf f_n dmu <= liminf ∫ f_n dmu (can be strict).
DCT: if |f_n| <= g ∈ L^1 and f_n → f a.e., then ∫ f_n dmu → ∫ f dmu (and ∫|f_n - f| dmu → 0).
MCT and Fatou apply to non-negative functions without integrability; DCT requires an L^1 dominator.
These theorems are the primary tools for exchanging limits and integrals in analysis, probability, and PDEs.

References

BookFolland — Real Analysis, 2nd ed. (1999), §2.2–§2.3
BookRudin — Real and Complex Analysis, 3rd ed. (1987), Theorems 1.26–1.34
WebsiteWikipedia — Dominated convergence theorem