Fatou's Lemma and the Dominated Convergence Theorem

Intuition Beginner

The monotone convergence theorem 02.07.04 says that when a non-negative function is approached pointwise from below by an increasing staircase, the integrals of the staircases climb up to the integral of the limit. This is the cleanest possible interaction between limits and integrals, but it is a strong hypothesis: the sequence has to be increasing. What if the sequence is just convergent — going up and down as it settles toward its limit?

Fatou's lemma (Fatou 1906) gives a one-sided answer for non-negative functions without any monotonicity hypothesis. If a sequence of non-negative measurable functions converges pointwise to a limit, the integral of the limit is at most the limit-inferior of the integrals. In symbols and prose: mass cannot appear from nowhere. The integrals can lose mass in the limit (think of a thin tall spike that moves off to infinity), but they cannot gain mass.

The classic picture is the running bump. Define $f_n$ on $[0,1]$ by $f_n(x)=n$ for $x$ in $[0,1/n]$ and $f_n(x)=0$ elsewhere. Every $f_n$ is a tall thin rectangle of height $n$ and base $1/n$. Its integral is exactly $1$. As $n$ grows, the rectangle shrinks horizontally and stretches vertically, and the pointwise limit is the zero function (every fixed $x$ eventually sits to the right of the shrinking interval). So the limit has integral $0$, but every approximating integral was $1$. The mass has disappeared into the singular point at the origin.

The dominated convergence theorem (Lebesgue 1908) rescues a two-sided answer when the sequence stays under an integrable envelope. If the absolute values $|f_n|$ all sit below a single non-negative integrable function $g$, then the limit of integrals equals the integral of the limit. The envelope $g$ acts like a ceiling that prevents the running-bump mass-escape: nothing can shoot off to infinity if everything is trapped below a finite-mass dome.

The one-sentence takeaway: integrals can lose mass under pointwise convergence (Fatou one-sided), but they cannot lose mass when a single integrable envelope dominates the sequence (Lebesgue two-sided).

Visual Beginner

Picture two scenarios. In the first, a sequence of tall thin spikes shrinks toward the origin: each spike has the same area, but the pointwise limit is flat zero. The integrals stay at $1$ while the limit-integral is $0$. Fatou's lemma says the integral of the limit (zero) is at most the limit of the integrals (one), and the inequality is strict because mass escaped.

In the second scenario, a sequence of bumps sits below a fixed tent-shaped envelope $g$ with finite area. As the bumps wiggle into their limit shape, no bump can shoot above the tent, so no mass can escape outside the tent's finite area. The dominated convergence theorem says the integrals converge to the integral of the limit.

Worked example Beginner

We use the running-bump example to see Fatou's lemma in action and to see why the dominated convergence theorem does not apply.

Step 1. Define $f_n$ on $[0,1]$ with Lebesgue measure by $f_n(x)=n$ when $0\le x\le 1/n$ and $f_n(x)=0$ when $1/n<x\le 1$. Each $f_n$ is a step function with two pieces.

Step 2. Compute the integral of each $f_n$. The rectangle has height $n$ and base $1/n$, so its area is $n\cdot (1/n)=1$. Every $f_n$ has integral exactly $1$, so the limit of these integrals is $1$.

Step 3. Compute the pointwise limit. Fix $x$ in $(0,1]$. Eventually $n>1/x$, which puts $x$ outside the interval $[0,1/n]$, so $f_n(x)=0$ for large $n$. The pointwise limit of $f_n(x)$ is $0$ for every $x>0$. At $x=0$, the values blow up to infinity, but the single point has measure $0$. So the pointwise limit is the zero function almost everywhere.

Step 4. The integral of the limit equals $0$, and the limit of the integrals equals $1$. Fatou's lemma says $0\le 1$, which is correct but a strict inequality. The dominated convergence theorem does not apply, because no single integrable envelope $g$ on $[0,1]$ can sit above every $f_n$ (any such $g$ would need $g(x)\ge n$ at some point near zero for every $n$, forcing $g$ to be unbounded near zero in a non-integrable way).

What this tells us: when mass can escape to a single point or out to infinity, integrals can shrink under pointwise convergence; the dominated convergence theorem prevents this by demanding a finite-mass envelope.

Check your understanding Beginner

Formal definition Intermediate+

Let $(X,\mathcal{M},\mu )$ be a measure space. We work with measurable functions $f_n,f:X\to \bar{\mathbb{R}}$ (the extended real line). The almost-everywhere modifier on a limit means equality outside a measurable null set; the inequalities below are read with the usual conventions $0\cdot \infty =0$ and $a-\infty =-\infty$ when subtraction is unambiguous.

Definition (limit-inferior of a sequence of functions). For measurable $f_n:X\to \bar{\mathbb{R}}$, the pointwise limit-inferior is $$\left(\underset{n\to \infty }{\mathrm{liminf}}{f}_n\right)(x)=\underset{n}{\sup }\underset{k\ge n}{\inf }{f}_k(x).$$ This is the pointwise supremum of the increasing sequence of measurable functions $g_n(x)={\inf }_{k\ge n}{f}_k(x)$, so ${\mathrm{liminf}}_n{f}_n$ is itself measurable.

Definition (uniform integrability). A family $\{f_{\alpha }\}$ of measurable functions on a finite-measure space $(X,\mathcal{M},\mu )$ is uniformly integrable when for every $\epsilon >0$ there exists $M>0$ such that for every $\alpha$, $${\int }_{\{|f_{\alpha }|>M\}}|f_{\alpha }|d\mu <\epsilon .$$ Equivalently, the family has uniformly absolutely continuous integrals: for every $\epsilon >0$ there exists $\delta >0$ such that for every $\alpha$ and every measurable $E$ with $\mu (E)<\delta$, ${\int }_E|f_{\alpha }|d\mu <\epsilon$ ^{[Folland §2.3]}.

Definition (convergence in measure). A sequence $f_n$ converges to $f$ in measure when for every $\epsilon >0$, $\mu (\{x:|f_n(x)-f(x)|>\epsilon \})\to 0$ as $n\to \infty$.

Counterexamples to common slips Intermediate+

• Fatou's lemma is not an equality. The running-bump $f_n=n{\chi }_{[0,1/n]}$ on $[0,1]$ has ${\mathrm{liminf}}_n{f}_n=0$ a.e., so $\int \mathrm{liminf}{f}_n=0$, while $\int f_n=1$ for every $n$ and ${\mathrm{liminf}}_n\int f_n=1$. The Fatou inequality $0\le 1$ is strict; the missing mass has escaped to the singular point at the origin.

• Fatou requires non-negativity (or a lower envelope). For $f_n=-{\chi }_{[n,n+1]}$ on the real line, ${\mathrm{liminf}}_n{f}_n=0$ a.e. (each fixed $x$ has $f_n(x)=0$ for large $n$) so $\int \mathrm{liminf}{f}_n=0$. But $\int f_n=-1$ for every $n$, so ${\mathrm{liminf}}_n\int f_n=-1<0=\int \mathrm{liminf}{f}_n$. The non-negativity hypothesis is what gives Fatou its direction.

• Dominated convergence requires a single $g$ for the whole sequence. The pointwise limit $|f_n|\le g_n$ with $g_n$ varying does not give DCT directly; one needs Pratt's lemma (Theorem 4 below) or a fixed envelope. The running-bump escapes precisely because no fixed $g$ dominates.

• DCT does not require pointwise convergence everywhere — almost everywhere is enough. Modification of $f_n$ on a null set does not change the integral, and the limit needs only hold a.e. The proof uses the fact that integrals are invariant under a.e. modification.

• Convergence in measure is strictly weaker than a.e. convergence. The typewriter sequence (the sequence of indicators of dyadic intervals scanning $[0,1]$ left-to-right, then halving) converges to $0$ in measure but not at any point of $[0,1]$. Vitali's convergence theorem (Theorem 3 below) handles such sequences as long as uniform integrability holds.

Key theorem with proof Intermediate+

Theorem (Lebesgue dominated convergence; Lebesgue 1908). Let $(X,\mathcal{M},\mu )$ be a measure space and let $f_n:X\to \bar{\mathbb{R}}$ be a sequence of measurable functions converging pointwise a.e. to a measurable $f:X\to \bar{\mathbb{R}}$. Suppose there exists $g\in L^1(X,\mu )$ with $|f_n|\le g$ a.e. for every $n$. Then $f\in L^1(X,\mu )$ and $${\int }_{X}fd\mu =\underset{n\to \infty }{\lim }{\int }_X{f}_{n}d\mu ,\underset{n\to \infty }{\lim }{\int }_X|f_n-f|d\mu =0.$$

Proof. By modification on a null set we may assume $|f_n|\le g$ and $f_n\to f$ pointwise everywhere on $X$. Then $|f(x)|\le g(x)$ everywhere as well, so $f$ is measurable (a.e.-pointwise limit of measurables) and $\int |f|\le \int g<\infty$, giving $f\in L^1$.

Step 1 (the upper Fatou). Apply Fatou's lemma (Theorem 1 below; the same lemma can be proved directly from MCT as in Theorem 1's proof) to the non-negative sequence $g+f_n\ge 0$, which has pointwise limit $g+f$: $${\int }_X(g+f)d\mu \le \underset{n}{\mathrm{liminf}}{\int }_X(g+f_n)d\mu ={\int }_{X}gd\mu +\underset{n}{\mathrm{liminf}}{\int }_X{f}_{n}d\mu .$$ Since $\int g<\infty$, we may subtract $\int g$ from both sides: $${\int }_{X}fd\mu \le \underset{n}{\mathrm{liminf}}{\int }_X{f}_{n}d\mu .$$

Step 2 (the reverse Fatou). Apply Fatou's lemma to the non-negative sequence $g-f_n\ge 0$, which has pointwise limit $g-f$: $${\int }_X(g-f)d\mu \le \underset{n}{\mathrm{liminf}}{\int }_X(g-f_n)d\mu ={\int }_{X}gd\mu -\underset{n}{\mathrm{limsup}}{\int }_X{f}_{n}d\mu .$$ Subtracting $\int g$ from both sides and multiplying by $-1$ (which reverses the inequality): $${\int }_{X}fd\mu \ge \underset{n}{\mathrm{limsup}}{\int }_X{f}_{n}d\mu .$$

Step 3 (sandwich). Putting Steps 1 and 2 together: $$\underset{n}{\mathrm{limsup}}{\int }_X{f}_{n}d\mu \le {\int }_{X}fd\mu \le \underset{n}{\mathrm{liminf}}{\int }_X{f}_{n}d\mu .$$ Since $\mathrm{liminf}\le \mathrm{limsup}$ always, every inequality is an equality, and ${\lim }_n\int f_n=\int f$.

Step 4 ($L^1$ convergence). Apply Steps 1-3 to the non-negative sequence $|f_n-f|\to 0$ pointwise, dominated by $2g\in L^1$. The conclusion ${\lim }_n\int |f_n-f|=\int 0=0$ is exactly the $L^1$-convergence claim.

$\square$

Bridge. The dominated convergence theorem builds toward 02.07.06 $L^p$ spaces, where the same two-Fatou-sandwich argument identifies convergence in $L^p$-norm with the combination of a.e. convergence and uniform integrability of the $p$-th powers. The central insight is that the dominating envelope $g$ acts as a finite-mass ceiling that traps the sequence and prevents mass-escape, and this is exactly the structural fact that Pratt 1960 generalises to variable dominators $g_n\to g$ in $L^1$. The foundational reason DCT follows from Fatou is the upper-and-lower bracketing: applying Fatou to $g+f_n$ gives the $\mathrm{liminf}$ direction and applying Fatou to $g-f_n$ gives the $\mathrm{limsup}$ direction; putting these together identifies the limit with the integral of the limit. The bridge is between the descriptive theory of pointwise convergence and the operational theory of integral-passage, and the pattern generalises to Banach-valued integration via Bochner DCT 02.11.04 and to probability theory where almost-sure convergence plus uniform integrability gives $L^1$ convergence, appears again in 26.02.07 pending as the load-bearing step in the continuity theorem for characteristic functions.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: full — Mathlib provides the central convergence theorems. MeasureTheory.lintegral_liminf_le is Fatou's lemma for ENNReal-valued measurable functions: for a sequence of measurable functions, the lower Lebesgue integral of the pointwise limit-inferior is at most the limit-inferior of the integrals. MeasureTheory.tendsto_lintegral_of_dominated_convergence is the dominated convergence theorem in the lower-Lebesgue (non-negative) setting; MeasureTheory.tendsto_integral_of_dominated_convergence is the Bochner (Banach-valued) version with domination by an L^1 function. Mathlib also provides MeasureTheory.UniformIntegrable and the Vitali convergence theorem variant for finite-measure spaces.

import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Function.UniformIntegrable

variable (X : Type*) [MeasurableSpace X]
variable (μ : MeasureTheory.Measure X)

abbrev CodexFatou (f : ℕ → X → ENNReal) : Prop :=
  MeasureTheory.lintegral μ (fun x => Filter.liminf (fun n => f n x) Filter.atTop)
    ≤ Filter.liminf (fun n => MeasureTheory.lintegral μ (f n)) Filter.atTop

abbrev CodexDominated {β : Type*} [NormedAddCommGroup β]
    (f : ℕ → X → β) (g : X → ℝ) : Prop :=
  ∀ n, ∀ᵐ x ∂μ, ‖f n x‖ ≤ g x

Advanced results Master

The advanced theory of the convergence theorems splits across five strands: the reverse-Fatou inequality with an upper envelope, Pratt's variable-dominator generalisation, Vitali convergence and uniform integrability, the de la Vallée Poussin criterion for uniform integrability, and the Bochner-valued extension of DCT to Banach-valued integration.

Theorem 1 (Fatou's lemma; Fatou 1906). Let $(X,\mathcal{M},\mu )$ be a measure space and let $\{f_n\}$ be a sequence of measurable functions $X\to [0,\infty ]$. Then $${\int }_X\underset{n\to \infty }{\mathrm{liminf}}{f}_{n}d\mu \le \underset{n\to \infty }{\mathrm{liminf}}{\int }_X{f}_{n}d\mu .$$ The inequality can be strict: the running bump $f_n=n{\chi }_{[0,1/n]}$ on $[0,1]$ has $\int \mathrm{liminf}=0$ and $\mathrm{liminf}\int =1$ ^{[Fatou 1906]}.

Proof. Define $g_n(x)={\inf }_{k\ge n}{f}_k(x)$. Each $g_n$ is measurable as a countable infimum. The sequence is monotone non-decreasing in $n$ with pointwise limit ${\mathrm{liminf}}_n{f}_n$. By monotone convergence applied to $\{g_n\}$: $${\int }_X\underset{n}{\mathrm{liminf}}{f}_{n}d\mu =\underset{n}{\lim }{\int }_X{g}_{n}d\mu .$$ For each $n$ and each $k\ge n$, $g_n\le f_k$ pointwise, so $\int g_n\le \int f_k$. Taking the infimum over $k\ge n$: $\int g_n\le {\inf }_{k\ge n}\int f_k$. Taking the limit as $n\to \infty$ (the left side increases, the right side is ${\mathrm{liminf}}_n\int f_n$ by definition): $$\underset{n}{\lim }\int g_{n}d\mu \le \underset{n}{\mathrm{liminf}}\int f_{n}d\mu .\square$$

Theorem 2 (reverse Fatou). Let $\{f_n\}$ be measurable functions $X\to \bar{\mathbb{R}}$ and suppose there exists a non-negative $g\in L^1(X,\mu )$ with $f_n\le g$ a.e. for every $n$. Then $$\underset{n\to \infty }{\mathrm{limsup}}{\int }_X{f}_{n}d\mu \le {\int }_X\underset{n\to \infty }{\mathrm{limsup}}{f}_{n}d\mu .$$

Proof. The sequence $g-f_n$ is non-negative and ${\mathrm{liminf}}_n(g-f_n)=g-{\mathrm{limsup}}_n{f}_n$. Apply Theorem 1: $${\int }_X(g-\underset{n}{\mathrm{limsup}}{f}_n)d\mu \le \underset{n}{\mathrm{liminf}}{\int }_X(g-f_n)d\mu ={\int }_{X}gd\mu -\underset{n}{\mathrm{limsup}}{\int }_X{f}_{n}d\mu .$$ Since $\int g<\infty$, subtracting gives ${\mathrm{limsup}}_n\int f_n\le \int {\mathrm{limsup}}_n{f}_n$. $\square$

Theorem 3 (Vitali convergence; Vitali 1907). Let $(X,\mathcal{M},\mu )$ be a measure space with $\mu (X)<\infty$. Let $\{f_n\}\subset L^1(X,\mu )$ converge to $f$ in measure. Then $f\in L^1$ and $\int f_n\to \int f$ if and only if $\{f_n\}$ is uniformly integrable.

The Vitali theorem strictly generalises DCT in the finite-measure case: every dominated sequence is uniformly integrable (with dominator $g$ supplying the bound), but there are uniformly integrable sequences that are not dominated by any single $L^1$ function (consider $f_n=n{\chi }_{[1/n,2/n]}$ on $[0,1]$, which is uniformly integrable because the integrals localise to shrinking intervals, but no $L^1$ envelope can dominate $n$ on $[1/n,2/n]$ for every $n$). Vitali's theorem is the standard tool in martingale theory: a martingale $\{X_n\}$ converges in $L^1$ if and only if it is uniformly integrable, the Doob $L^1$-convergence theorem ^{[Vitali 1907]}.

Theorem 4 (Pratt's lemma; Pratt 1960). Let $\{f_n\},\{g_n\},\{h_n\}$ be sequences of measurable functions on $(X,\mathcal{M},\mu )$ with $f_n\to f$, $g_n\to g$, $h_n\to h$ a.e., $g_n\le f_n\le h_n$ a.e., $g_n,h_n\in L^1$, and $\int g_n\to \int g$, $\int h_n\to \int h$ with $g,h\in L^1$. Then $f\in L^1$ and $\int f_n\to \int f$.

The special case $g_n=-h_n$ with $h_n=g$ constant recovers DCT. The general case allows the dominators to vary, which is essential in applications to stochastic processes (where the natural dominators are martingale increments rather than a single envelope) and to weak convergence (where the dominators converge in $L^1$ to a limit) ^{[Pratt 1960]}.

Proof. Apply Fatou to $f_n-g_n\ge 0$ (pointwise limit $f-g$): $$\int (f-g)d\mu \le \mathrm{liminf}\int (f_n-g_n)=\mathrm{liminf}\int f_n-\lim \int g_n=\mathrm{liminf}\int f_n-\int g.$$ So $\int f\le \mathrm{liminf}\int f_n$.

Apply Fatou to $h_n-f_n\ge 0$ (pointwise limit $h-f$): $$\int (h-f)\le \mathrm{liminf}\int (h_n-f_n)=\lim \int h_n-\mathrm{limsup}\int f_n=\int h-\mathrm{limsup}\int f_n.$$ So $\mathrm{limsup}\int f_n\le \int f$. Combining, $\lim \int f_n=\int f$. $\square$

Theorem 5 (de la Vallée Poussin criterion; de la Vallée Poussin 1915). A family $\{f_{\alpha }\}$ in $L^1(X,\mu )$ on a finite-measure space is uniformly integrable if and only if there exists an increasing convex function $\phi :[0,\infty )\to [0,\infty )$ with $\phi (t)/t\to \infty$ as $t\to \infty$ such that $$\underset{\alpha }{\sup }{\int }_X\phi (|f_{\alpha }|)d\mu <\infty .$$

The standard choices for $\phi$ are $\phi (t)=t^p$ for $p>1$ (which gives the $L^p$-bounded $⟹$ uniformly integrable implication, the key estimate behind $L^p$-compactness in the weak topology for $p>1$), $\phi (t)=t\ln (1+t)$ (which appears in Orlicz-space theory), and $\phi (t)=t{\log }_{+}(t)$ (which is the borderline at $p=1$ where uniform integrability fails to follow from $L^1$-boundedness alone) ^{[deLaValleePoussin 1915]}.

Theorem 6 (Dunford-Pettis theorem; Dunford-Pettis 1940). On a finite-measure space, a subset $K\subseteq L^1(X,\mu )$ is relatively weakly compact (precompact in the weak topology of $L^1$) if and only if it is bounded and uniformly integrable.

This is the analytic content of "uniform integrability is the right finite-measure analogue of weak compactness." It is the standard tool in the calculus of variations (semicontinuity of integrals along weakly convergent sequences requires Fatou + uniform integrability), in PDE (Galerkin approximations and existence of weak solutions), and in probability (tightness of measures + the Dunford-Pettis criterion characterises convergence in distribution against bounded continuous test functions).

Theorem 7 (Bochner dominated convergence; Bochner 1933). Let $E$ be a Banach space, $\{f_n\}$ strongly measurable $X\to E$ converging a.e. to $f$, and suppose there exists $g\in L^1(X,\mu ,\mathbb{R})$ with $\Vert f_n(x){\Vert }_E\le g(x)$ a.e. for every $n$. Then $f$ is Bochner integrable and $$\underset{n}{\lim }{\Vert {\int }_X{f}_{n}d\mu -{\int }_{X}fd\mu \Vert }_E=0,\underset{n}{\lim }{\int }_X\Vert f_n-f{\Vert }_{E}d\mu =0.$$

The Bochner DCT is the load-bearing theorem in the theory of Banach-valued random variables and in the spectral theorem for unbounded operators (where eigenfunction expansions involve Bochner integrals against the spectral measure). The proof follows the scalar DCT with $\Vert f_n-f{\Vert }_E$ in place of $|f_n-f|$ and the scalar DCT applied to the real-valued $\Vert f_n-f{\Vert }_E$ ^{[Bochner 1933]}.

Theorem 8 (continuity-under-integral with parameter). Let $f:X\times T\to \mathbb{R}$ where $T\subseteq {\mathbb{R}}^k$ is open. Suppose $x\mapsto f(x,t)$ is measurable for each $t$, $t\mapsto f(x,t)$ is continuous at $t_0$ for a.e. $x$, and there exists $g\in L^1(X,\mu )$ with $|f(x,t)|\le g(x)$ for a.e. $x$ and every $t$ in a neighbourhood of $t_0$. Then $F(t)={\int }_{X}f(x,t)d\mu (x)$ is continuous at $t_0$.

This is the parameter-integral continuity theorem, the workhorse for the Fourier transform on $L^1$ (continuity of $\hat{f}$ at every $\xi$), for the Laplace transform on $L^1({\mathbb{R}}_{\ge 0})$ (continuity on $\{\mathrm{Re}(s)>0\}$), and for the Mellin transform of $L^1$-functions of compact support.

Synthesis. The dominated convergence theorem is the foundational reason that twentieth-century analysis can interchange limits and integrals without case-by-case hand-checking. The central insight is the upper-and-lower bracketing of $\lim \int f_n$ by applying Fatou to $g+f_n$ and $g-f_n$, which identifies the integral of the limit with the limit of integrals whenever a single integrable envelope traps the sequence; this is exactly the structural fact that generalises to Pratt's variable-dominator setting, to Vitali's uniform-integrability setting on finite-measure spaces, and to the Bochner-valued setting for Banach-space-valued integrals.

The structural pattern generalises through three escalations. First, scalar DCT extends to Banach-valued DCT via Bochner 1933: the same Fatou-sandwich proof carries over with $\Vert f_n-f{\Vert }_E$ in place of $|f_n-f|$ and the scalar DCT applied to the real-valued norm-difference. Second, the constant-envelope hypothesis loosens to variable envelopes via Pratt 1960: as long as the bracketing functions $g_n,h_n$ converge in $L^1$ to their limits, the Fatou applications still close. Third, the finite-measure case loosens the domination hypothesis to uniform integrability via Vitali 1907 and the Dunford-Pettis criterion: a sequence whose integrals are uniformly absolutely continuous (the precise rephrasing of uniform integrability) converges in $L^1$ to its in-measure limit, the foundational fact behind the $L^1$-weak-compactness criterion. The bridge is between the descriptive theory (convergence modes — pointwise, in measure, in $L^p$) and the operational theory (integral-passage under each mode), and putting these together identifies the integration theory with the right convergence calculus on $L^1$, the pattern recurring in the martingale convergence theorems (Doob 1953), in the calculus of variations (semicontinuity of integral functionals under weak convergence), and in modern PDE (Galerkin existence proofs and weak solutions via Fatou plus uniform integrability).

Full proof set Master

Proposition 1 (Fatou with arbitrary lower envelope). Let $\{f_n\}$ be measurable $X\to \bar{\mathbb{R}}$ and suppose there exists $h\in L^1(X,\mu )$ with $f_n\ge h$ a.e. for every $n$. Then $${\int }_X\underset{n}{\mathrm{liminf}}{f}_{n}d\mu \le \underset{n}{\mathrm{liminf}}{\int }_X{f}_{n}d\mu .$$

Proof. The sequence $f_n-h$ is non-negative and a.e.-measurable. Apply Theorem 1 to $\{f_n-h\}$: $${\int }_X\underset{n}{\mathrm{liminf}}(f_n-h)d\mu \le \underset{n}{\mathrm{liminf}}{\int }_X(f_n-h)d\mu .$$ The left side equals ${\int }_X({\mathrm{liminf}}_n{f}_n-h)d\mu ={\int }_X{\mathrm{liminf}}_n{f}_{n}d\mu -{\int }_{X}hd\mu$ (using $\int h\in \mathbb{R}$). The right side equals ${\mathrm{liminf}}_n{\int }_X{f}_{n}d\mu -{\int }_{X}hd\mu$. Adding $\int h$ to both sides gives the conclusion.

$\square$

Proposition 2 (DCT implies $L^p$ convergence for $1\le p<\infty$). Let $\{f_n\}$ converge a.e. to $f$ with $|f_n|\le g$ a.e. for $g\in L^p$, $1\le p<\infty$. Then $\Vert f_n-f{\Vert }_{L^p}\to 0$.

Proof. Apply DCT to the non-negative sequence $|f_n-f{|}^p$, which converges to $0$ a.e. and is dominated by $(|f_n|+|f|{)}^p\le (2g{)}^p=2^p{g}^p\in L^1$ (using $|f|\le g$ a.e. from passing to the limit in $|f_n|\le g$). The DCT conclusion gives $\int |f_n-f{|}^p\to 0$, which is the $L^p$-convergence claim.

$\square$

Proposition 3 (Egorov's theorem feeds into DCT). Let $\mu (X)<\infty$ and $f_n\to f$ a.e. Then for every $\delta >0$ there is a measurable $E_{\delta }$ with $\mu (X\setminus E_{\delta })<\delta$ on which $f_n\to f$ uniformly.

Proof. This is Egorov's theorem from 02.07.03. The relevance here is that Egorov plus DCT gives a constructive route to convergence: uniform convergence on $E_{\delta }$ gives ${\int }_{E_{\delta }}{f}_n\to {\int }_{E_{\delta }}f$ directly (uniform convergence of bounded sequences on a finite-measure set), and the tail $X\setminus E_{\delta }$ has small measure, contributing at most ${\int }_{X\setminus E_{\delta }}gd\mu$, which is small by absolute continuity of the integral [02.07.04 Theorem 4] if a dominator $g$ is in hand.

$\square$

Proposition 4 (Pratt's lemma reduces to DCT via re-bracketing). The conclusion of Pratt's lemma (Theorem 4) can also be obtained by the dominated convergence theorem applied to the sequence ${\tilde{f}}_n=f_n+g_n^{-}+h_n^{-}$, where $g_n^{-},h_n^{-}$ are non-negative correction terms making the bracketing two-sided around $0$.

Proof. Set ${\varphi }_n=(h_n-g_n)/2$ and ${\psi }_n=(h_n+g_n)/2$, so $h_n={\varphi }_n+{\psi }_n$ and $g_n={\psi }_n-{\varphi }_n$. The hypothesis $g_n\le f_n\le h_n$ becomes $|f_n-{\psi }_n|\le {\varphi }_n$. With ${\varphi }_n\to \varphi$ in $L^1$ (consequence of $h_n-g_n\to h-g$ in $L^1$) and ${\psi }_n\to \psi$ in $L^1$, the sequence $f_n-{\psi }_n$ is bracketed by $\pm {\varphi }_n$, and a direct DCT-with-variable-dominator argument closes. The technical step is showing that an $L^1$-convergent sequence of dominators behaves like a fixed dominator for DCT purposes; this is exactly the content of Pratt's original argument.

$\square$

Proposition 5 (uniform integrability is closed under $L^1$ convergence). If $f_n\to f$ in $L^1$ on a finite-measure space, then $\{f_n\}$ is uniformly integrable.

Proof. Given $\epsilon >0$, choose $N$ such that $\Vert f_n-f{\Vert }_{L^1}<\epsilon /2$ for $n\ge N$. The single function $f$ is uniformly integrable (in the singleton sense): there is ${\delta }_1>0$ such that $\mu (E)<{\delta }_1⟹{\int }_E|f|<\epsilon /2$ (absolute continuity of the integral). Similarly the finite family $\{f_1,\dots ,f_{N-1}\}$ is uniformly integrable: choose ${\delta }_2>0$ such that $\mu (E)<{\delta }_2⟹{\int }_E|f_k|<\epsilon$ for $k<N$. For $n\ge N$ and $\mu (E)<{\delta }_1$: $${\int }_E|f_n|\le {\int }_E|f_n-f|+{\int }_E|f|\le \Vert f_n-f{\Vert }_{L^1}+\epsilon /2<\epsilon .$$ Take $\delta =\min ({\delta }_1,{\delta }_2)$. For any $n$ and any $E$ with $\mu (E)<\delta$, ${\int }_E|f_n|<\epsilon$.

$\square$

Proposition 6 (Dunford-Pettis via uniform integrability and Banach-Alaoglu). A bounded uniformly integrable set $K\subseteq L^1(X,\mu )$ on a finite-measure space has weak limits of every sequence in $L^1$.

Proof. Banach-Alaoglu applied to the dual pairing $(L^1,L^{\infty })$ gives weak-* compactness of bounded sets in $(L^{\infty }{)}^{\ast }\supseteq L^1$. The uniform integrability hypothesis is what ensures that the weak-* limit lies in $L^1$ rather than in the strictly larger dual $(L^{\infty }{)}^{\ast }$, which contains finitely additive set functions. The technical step is showing that uniform integrability prevents the loss of $L^1$-mass at infinity in the weak-* compactification, exactly the content of the Dunford-Pettis theorem [Theorem 6].

$\square$

Proposition 7 (Scheffé's lemma; Scheffé 1947). If $f_n,f\in L^1(X,\mu )$ are non-negative, $f_n\to f$ a.e., and $\int f_n\to \int f$, then $\int |f_n-f|\to 0$ (so $f_n\to f$ in $L^1$).

Proof. The non-negative sequence $(f-f_n{)}^{+}=\max (f-f_n,0)$ converges a.e. to $0$ and is dominated by $f$, which is integrable. By DCT, $\int (f-f_n{)}^{+}d\mu \to 0$. Write $|f_n-f|=(f_n-f)+2(f-f_n{)}^{+}$. Integrating: $$\int |f_n-f|d\mu =\int (f_n-f)d\mu +2\int (f-f_n{)}^{+}d\mu =\left(\int f_n-\int f\right)+2\int (f-f_n{)}^{+}.$$ Both pieces go to $0$: the first by hypothesis $\int f_n\to \int f$, the second by the DCT bound above. Hence $\int |f_n-f|\to 0$. $\square$

This proposition is the converse-style companion to DCT: instead of assuming a dominator and deducing $L^1$ convergence from a.e. convergence, Scheffé's lemma assumes equality of integral limits and deduces $L^1$ convergence from a.e. convergence. The proposition is the standard tool in probability theory for converting convergence of densities to convergence in total variation distance between probability measures.

Proposition 8 (DCT and the Lévy continuity theorem). Let $X_n$ be random variables on a probability space with characteristic functions ${\varphi }_n(\xi )=\mathbb{E}[e^{i\xi X_n}]$. If $X_n\to X$ in distribution, then ${\varphi }_n(\xi )\to \varphi (\xi )$ for every $\xi$; conversely, if ${\varphi }_n(\xi )\to \psi (\xi )$ for every $\xi$ and $\psi$ is continuous at $0$, then $\psi$ is the characteristic function of some random variable $X$ and $X_n\to X$ in distribution.

Proof of the forward direction. The family of bounded continuous functions $\{e^{i\xi \cdot }{\}}_{\xi \in \mathbb{R}}$ is the test class for convergence in distribution (Portmanteau theorem). For each fixed $\xi$, the function $x\mapsto e^{i\xi x}$ is bounded continuous, so $X_n\to X$ in distribution gives $\mathbb{E}[e^{i\xi X_n}]\to \mathbb{E}[e^{i\xi X}]$, i.e., ${\varphi }_n(\xi )\to \varphi (\xi )$. The DCT enters in the converse via the proof that $\psi$ is a characteristic function — one shows $\psi$ extends to a positive-definite continuous function and applies Bochner's theorem (characteristic-function-characterisation theorem). $\square$

The Lévy continuity theorem is the analytic backbone of the central limit theorem 26.02.07 pending: the convergence of suitably normalised sums of i.i.d. random variables in distribution to the Gaussian is proved by showing the characteristic functions converge pointwise to $e^{-{\xi }^2/2}$, then applying Lévy continuity to lift this pointwise convergence to convergence in distribution.

Connections Master

• Lebesgue integral construction and the monotone convergence theorem 02.07.04. The direct prerequisite. MCT is the primitive limit theorem; Fatou's lemma is the $\mathrm{liminf}$ version obtained by applying MCT to the increasing infimum-tail sequence $g_n={\inf }_{k\ge n}{f}_k$; DCT is obtained by applying Fatou to $g-f_n$ and $g+f_n$ to bracket the limit from above and below. The chain of inferences MCT $\to$ Fatou $\to$ DCT is the load-bearing skeleton of the convergence theorems.

• Measurable functions, simple functions, Egorov's theorem, and Lusin's theorem 02.07.03. Provides the framework on which convergence is built: pointwise a.e. convergence of measurable functions, Egorov's theorem refining a.e. convergence to uniform convergence outside a small set, and Lusin's theorem refining measurable functions to continuous functions outside a small set. Egorov plus DCT gives a constructive route to the convergence-of-integrals conclusion on finite-measure spaces.

• $L^p$ spaces, Hölder, Minkowski, and Riesz-Fischer completeness 02.07.06. Built on the convergence theorems of this unit. DCT applied to $|f_n-f{|}^p$ gives $L^p$ convergence from a.e. convergence plus $L^p$ domination; the Riesz-Fischer completeness theorem for $L^p$ is a direct application of DCT to Cauchy sequences. The duality theory of $L^p$ for $1\le p<\infty$ relies on the Dunford-Pettis theorem (Theorem 6) in the $p=1$ case.

• Banach spaces, completeness, and Bochner integration 02.11.04. The Bochner integral for Banach-valued functions extends DCT to vector-valued settings (Theorem 7). The Riesz-Fischer completeness of $L^p$ identifies $L^p$ as a Banach space; the Bochner integral is the canonical example of a Banach-valued integration theory; and DCT is the workhorse for both the construction (limits of simple-function approximations) and the operational use (interchange of limits and integrals) of Banach-valued integration.

• Central limit theorem and characteristic-function continuity 26.02.07 pending. DCT is essential in the continuity theorem for characteristic functions in probability theory (Lévy's continuity theorem): if random variables $X_n$ converge in distribution to $X$, the characteristic functions ${\varphi }_{X_n}(\xi )=\mathbb{E}[e^{i\xi X_n}]$ converge to ${\varphi }_X(\xi )$ pointwise, and the converse direction uses DCT on the bounded continuous family $\{e^{i\xi X}\}$ to lift pointwise convergence of characteristic functions to convergence in distribution. The bridge between analytic limit theorems (DCT) and probabilistic limit theorems (the central limit theorem) runs through characteristic functions.

Historical & philosophical context Master

Fatou's 1906 Acta Mathematica paper ^{[Fatou 1906]} introduced the lemma $\int \mathrm{liminf}{f}_n\le \mathrm{liminf}\int f_n$ for non-negative measurable functions. Fatou's main result in the paper was about boundary behaviour of bounded analytic functions in the unit disc — the Fatou theorem on radial limits — and the lemma now bearing his name was a technical tool deployed in service of the headline result. The extraction of Fatou's lemma as a stand-alone measure-theoretic theorem and its identification as a primitive limit-passage statement is due to the modern textbook tradition (Halmos 1950, Rudin 1966, Folland 1984).

Lebesgue's 1908 second edition of Leçons sur l'intégration ^{[Lebesgue 1908]} contains the dominated convergence theorem in its modern form. Lebesgue's original statement was for a sequence of measurable functions on a bounded interval with a uniform bound $|f_n|\le M$ (the bounded convergence theorem); the extension to a general integrable dominator $g$ was the work of the early twentieth-century French school, including Lebesgue's own iterations and the contributions of Riesz, Borel, and Vitali.

Vitali's 1907 Rendiconti del Circolo Matematico di Palermo paper ^{[Vitali 1907]} introduced the uniform integrability condition and proved that, on a finite-measure space, convergence in measure plus uniform integrability is equivalent to convergence of integrals. Vitali's framing was in terms of "integrable in series" — a quantitative absolute-continuity condition on the integrals over small-measure sets — and his theorem was originally a tool for studying term-by-term integration of trigonometric series, the same kind of problem motivating Fatou's lemma.

De la Vallée Poussin's 1915 Transactions of the American Mathematical Society paper ^{[deLaValleePoussin 1915]} gave the criterion characterising uniform integrability as the existence of an increasing convex superlinear $\phi$ with ${\sup }_n\int \phi (|f_n|)d\mu <\infty$. De la Vallée Poussin's framing connected uniform integrability to the modulus of continuity of the integral and to the Orlicz-space framework (later developed by Orlicz 1932 Bull. Acad. Polon. Sci. and Krasnoselskii-Rutickii 1961 monograph).

Pratt's 1960 Transactions of the American Mathematical Society paper ^{[Pratt 1960]} gave the variable-dominator generalisation of DCT. Pratt's motivation was statistical decision theory, where the natural dominators are not fixed envelopes but sequences converging in $L^1$ to a limit. Pratt's lemma is the standard tool in the modern theory of stochastic integration (Itô's formula derivation), in the calculus of variations (passing to the limit in integral functionals), and in the theory of weak solutions to PDE (Galerkin approximation arguments where the dominators are partial sums of an expansion).

Bochner's 1933 Fundamenta Mathematicae paper ^{[Bochner 1933]} extended DCT to Banach-valued functions. The Bochner construction follows the same three-step pattern as the scalar Lebesgue integral, with simple functions taking values in a Banach space and the integrability condition $\int \Vert f{\Vert }_{E}d\mu <\infty$ replacing the scalar finiteness condition. The Bochner DCT is the foundation of modern probability theory of Banach-valued random variables (Ledoux-Talagrand 1991 Probability in Banach Spaces), of spectral theory of unbounded operators (Dunford-Schwartz 1958-71 Linear Operators), and of harmonic analysis on non-commutative groups.

The Dunford-Pettis theorem characterising relatively weakly compact subsets of $L^1$ as the bounded uniformly integrable subsets (Dunford 1939, Pettis 1938, Dunford-Pettis 1940 Trans. AMS 47) closes a loop with Vitali 1907: uniform integrability is both the right finite-measure analogue of weak compactness in $L^1$ and the right condition for the convergence-of-integrals conclusion. The theorem is the analytic content of "uniform integrability = weak compactness in $L^1$" and underlies the modern treatment of weak convergence in probability (Billingsley 1968) and in the calculus of variations (Dacorogna 2008).

The structural story of the convergence theorems is a sixty-year arc: Fatou 1906 (lemma in trigonometric series context) → Beppo Levi 1906 (MCT) → Vitali 1907 (uniform integrability) → Lebesgue 1908 (DCT in monograph form) → de la Vallée Poussin 1915 (criterion for UI) → Bochner 1933 (Banach-valued) → Dunford-Pettis 1940 (weak compactness in $L^1$) → Pratt 1960 (variable dominators). Each step extends the previous to a more general setting while preserving the load-bearing two-Fatou-sandwich structure, and the result is a body of theorems that organises every convergence-of-integrals question under one of three patterns (monotone, dominated, uniformly integrable).

Bibliography Master

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