All Posts
Mathematics

The Dominated Convergence Theorem: When Can We Swap Limits and Integrals?

We prove the Dominated Convergence Theorem — the central result of Lebesgue integration theory that gives sharp conditions for interchanging limits and integrals — and compare it to weaker results like the Monotone Convergence Theorem.

Measure TheoryAnalysis

The Theorem

Dominated Convergence Theorem (Lebesgue)

Let (X,M,μ)(X, \mathcal{M}, \mu) be a measure space and let fn:X→Rf_n: X \to \mathbb{R} be measurable functions such that:

  1. fn(x)→f(x)f_n(x) \to f(x) pointwise a.e.
  2. There exists an integrable function g≥0g \geq 0 with ∣fn(x)∣≤g(x)|f_n(x)| \leq g(x) a.e. for all nn.

Then ff is integrable and:

lim⁡n→∞∫Xfn dμ=∫Xf dμ\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu

In short: if a pointwise convergent sequence of functions is dominated by a single integrable function, we may pass the limit inside the integral.

Why We Need It

The fundamental question of interchanging limits and integrals arises everywhere in analysis:

lim⁡n→∞∫fn=∫lim⁡n→∞fn???\lim_{n \to \infty} \int f_n = \int \lim_{n \to \infty} f_n \quad \text{???}

Without conditions, this can fail spectacularly:

A Counterexample

Define fn:[0,1]→Rf_n: [0,1] \to \mathbb{R} by fn=n⋅1(0,1/n]f_n = n \cdot \mathbf{1}_{(0, 1/n]}. Then:

  • fn(x)→0f_n(x) \to 0 for all x∈[0,1]x \in [0,1] (pointwise).
  • ∫01fn dx=n⋅1n=1\int_0^1 f_n \, dx = n \cdot \frac{1}{n} = 1 for all nn.

So lim⁡∫fn=1≠0=∫lim⁡fn\lim \int f_n = 1 \neq 0 = \int \lim f_n.

The problem: fnf_n concentrates mass. No single integrable function dominates all fnf_n (since sup⁡nfn(x)=∞\sup_n f_n(x) = \infty near 00).

The Domination Condition

The hypothesis ∣fn∣≤g|f_n| \leq g with g∈L1g \in L^1 is the key. It prevents the sequence from "escaping to infinity" or "concentrating mass" — the two main ways the interchange can fail.

Think of gg as a "safety net." As long as each fnf_n stays below gg (in absolute value), and gg has finite integral, the convergence is well-behaved.

Proof

The proof uses Fatou's Lemma as a stepping stone.

Fatou's Lemma

Fatou's Lemma. If fn≥0f_n \geq 0 are measurable, then:

∫Xlim inf⁡n→∞fn dμ≤lim inf⁡n→∞∫Xfn dμ\int_X \liminf_{n \to \infty} f_n \, d\mu \leq \liminf_{n \to \infty} \int_X f_n \, d\mu

Fatou's Lemma requires no domination — only non-negativity. The inequality can be strict (as our counterexample shows). The DCT upgrades this to an equality.

Proof of DCT

Proof.

Since ∣fn∣≤g|f_n| \leq g a.e. and fn→ff_n \to f a.e., we have ∣f∣≤g|f| \leq g a.e., so f∈L1f \in L^1.

Upper bound. The functions g−fn≥0g - f_n \geq 0 a.e. By Fatou's Lemma:

∫(g−f) dμ≤lim inf⁡∫(g−fn) dμ\int (g - f)\, d\mu \leq \liminf \int (g - f_n)\, d\mu

∫g−∫f≤∫g−lim sup⁡∫fn\int g - \int f \leq \int g - \limsup \int f_n

lim sup⁡∫fn≤∫f\limsup \int f_n \leq \int f

Lower bound. The functions g+fn≥0g + f_n \geq 0 a.e. By Fatou's Lemma:

∫(g+f) dμ≤lim inf⁡∫(g+fn) dμ\int (g + f)\, d\mu \leq \liminf \int (g + f_n)\, d\mu

∫g+∫f≤∫g+lim inf⁡∫fn\int g + \int f \leq \int g + \liminf \int f_n

∫f≤lim inf⁡∫fn\int f \leq \liminf \int f_n

Conclusion. Combining both:

∫f≤lim inf⁡∫fn≤lim sup⁡∫fn≤∫f\int f \leq \liminf \int f_n \leq \limsup \int f_n \leq \int f

Therefore lim⁡∫fn=∫f\lim \int f_n = \int f. □\square

Comparison of Convergence Theorems

Monotone Convergence

0≤f1≤f2≤⋯0 \leq f_1 \leq f_2 \leq \cdots

∫lim⁡fn=lim⁡∫fn\int \lim f_n = \lim \int f_n

(Limit may be +∞+\infty)

Fatou's Lemma

fn≥0f_n \geq 0

∫lim inf⁡fn≤lim inf⁡∫fn\int \liminf f_n \leq \liminf \int f_n

(Inequality only)

Dominated Convergence

∣fn∣≤g∈L1|f_n| \leq g \in L^1

∫lim⁡fn=lim⁡∫fn\int \lim f_n = \lim \int f_n

(Exact equality)

The DCT is the most powerful for applications, but requires the strongest hypothesis.

The LpL^p Version

LpL^p Dominated Convergence. If fn→ff_n \to f a.e., ∣fn∣≤g|f_n| \leq g a.e. with g∈Lpg \in L^p (1≤p<∞1 \leq p < \infty), then fn→ff_n \to f in LpL^p:

lim⁡n→∞∫∣fn−f∣p dμ=0\lim_{n \to \infty} \int |f_n - f|^p \, d\mu = 0

This is stronger than just ∫fn→∫f\int f_n \to \int f — it gives convergence in the LpL^p norm.

Applications

Differentiating Under the Integral Sign

Leibniz's Rule. Let F(θ)=∫Xf(x,θ) dμ(x)F(\theta) = \int_X f(x, \theta)\, d\mu(x). If ∂f/∂θ\partial f / \partial \theta exists and is dominated by an integrable function:

∣∂f∂θ(x,θ)∣≤g(x)for all θ\left|\frac{\partial f}{\partial \theta}(x, \theta)\right| \leq g(x) \quad \text{for all } \theta

then:

F′(θ)=∫X∂f∂θ(x,θ) dμ(x)F'(\theta) = \int_X \frac{\partial f}{\partial \theta}(x, \theta)\, d\mu(x)

Proof. Write the derivative as a limit of difference quotients and apply the DCT. □\square

This is used constantly in probability theory, physics, and PDE theory.

Moment Generating Functions

For a random variable XX with E[etX]<∞E[e^{tX}] < \infty for ∣t∣<c|t| < c, the moment generating function M(t)=E[etX]M(t) = E[e^{tX}] is infinitely differentiable on (−c,c)(-c, c), and:

M(n)(t)=E[XnetX]M^{(n)}(t) = E[X^n e^{tX}]

by iterated application of the DCT.

Fourier Transform Continuity

If f∈L1(Rn)f \in L^1(\mathbb{R}^n), the Fourier transform f^(ξ)=∫f(x)e−2πix⋅ξ dx\hat{f}(\xi) = \int f(x) e^{-2\pi i x \cdot \xi}\, dx is continuous, because ∣f(x)e−2πix⋅ξ∣≤∣f(x)∣|f(x)e^{-2\pi i x \cdot \xi}| \leq |f(x)| and the DCT applies to ξ→ξ0\xi \to \xi_0.

Interchange of Summation and Integration

Since ∑k=1nfk=Sn\sum_{k=1}^n f_k = S_n is a sequence of partial sums, the DCT gives conditions for:

∫∑k=1∞fk=∑k=1∞∫fk\int \sum_{k=1}^{\infty} f_k = \sum_{k=1}^{\infty} \int f_k

Specifically, if ∑∣fk∣≤g∈L1\sum |f_k| \leq g \in L^1, the interchange is valid.

Vitali's Convergence Theorem

A natural question: can the domination condition be weakened? Yes — if you replace it with uniform integrability.

Vitali's Convergence Theorem. On a finite measure space, if fn→ff_n \to f in measure and {fn}\{f_n\} is uniformly integrable, then:

lim⁡n→∞∫fn dμ=∫f dμ\lim_{n \to \infty} \int f_n \, d\mu = \int f \, d\mu

Uniform integrability is the precise condition needed — domination implies uniform integrability but not conversely.

Historical Context

Henri Lebesgue introduced both the Lebesgue integral and the Dominated Convergence Theorem in his 1904 dissertation "Intégrale, longueur, aire." The DCT was one of the main motivations for the Lebesgue theory — the Riemann integral lacks a comparably powerful convergence theorem.

The Monotone Convergence Theorem (Beppo Levi, 1906) and Fatou's Lemma (Pierre Fatou, 1906) completed the trio of fundamental convergence results.

Summary

fn→f a.e.,∣fn∣≤g∈L1⇓ DCT∫fn→∫fKey applications:Differentiation under the integralFourier analysis, probability, PDEsInterchange of ∑ and ∫\begin{aligned} &f_n \to f \text{ a.e.}, \quad |f_n| \leq g \in L^1 \\[6pt] &\Downarrow \text{ DCT} \\[6pt] &\int f_n \to \int f \\[8pt] &\text{Key applications:} \\[4pt] &\quad \text{Differentiation under the integral} \\[4pt] &\quad \text{Fourier analysis, probability, PDEs} \\[4pt] &\quad \text{Interchange of }\sum \text{ and } \int \end{aligned}

References

Back to all posts