Dominated Convergence Theorem

Statement

Let (X, A, μ) be a measure space and (f_n) a sequence of measurable functions X → ℝ. Suppose:

• Pointwise convergence: f_n(x) → f(x) for μ-almost every x ∈ X.
• Dominator: There exists a measurable g : X → [0, ∞] with ∫g dμ < ∞ and |f_n(x)| ≤ g(x) for all n and almost every x.

Then f is integrable, and:

∫ fₙ dμ → ∫ f dμ
∫ |fₙ - f| dμ → 0       (an even stronger conclusion)

The pointwise limit f inherits measurability automatically (limits of measurable functions are measurable) and integrability follows because |f| ≤ g almost everywhere.

Worked example — sin(nx)/n on [0, 1]

Let f_n(x) = sin(nx)/n on [0, 1]. Step through DCT:

1. Pointwise convergence. For any x ∈ [0, 1], |sin(nx)/n| ≤ 1/n → 0. So f_n(x) → 0 = f(x) pointwise.
2. Dominator. Take g(x) = 1. Then |f_n(x)| = |sin(nx)/n| ≤ 1/n ≤ 1 = g(x), and ∫₀¹ 1 dx = 1 < ∞.
3. Conclusion. By DCT, ∫₀¹ f_n dx → ∫₀¹ 0 dx = 0.

Verify directly: ∫₀¹ sin(nx)/n dx = [−cos(nx)/n²]₀¹ = (1 − cos n)/n² → 0 as n → ∞. ✓

This particular example is easy enough that we could compute the integrals exactly. The power of DCT is that you almost never have to — pointwise + dominator is a single fact to check, and the integrals follow automatically.

Counterexample — when the dominator fails

Sliding rectangles on [0, 1]:

fₙ(x) = n · 1_{(0, 1/n]}(x)
       = n  if 0 < x ≤ 1/n
         0  otherwise

Pointwise: for any x > 0, fₙ(x) = 0 for n > 1/x. So fₙ → 0.

∫fₙ dx = n · (1/n) = 1     for every n.

So lim ∫fₙ = 1 ≠ 0 = ∫lim fₙ.

Why does this not contradict DCT? Because no integrable dominator exists. Any g satisfying |f_n| ≤ g would need g(x) ≥ n for some n with 0 < x ≤ 1/n. Taking n = ⌈1/x⌉ gives g(x) ≥ 1/x, which is not integrable on (0, 1]. So DCT's hypotheses fail; its conclusion correctly doesn't apply.

Spreading rectangles tell the same story going the other way:

fₙ(x) = (1/n) · 1_{[0, n]}(x)   on [0, ∞)

Pointwise: fₙ(x) → 0 (eventually 1/n < ε for any ε).
∫fₙ dx = (1/n) · n = 1 for every n.
Mass spreads out instead of concentrating — but again no integrable dominator.

Proof sketch — Fatou's lemma is the engine

Apply Fatou's lemma to the non-negative sequence g + f_n ≥ 0 (since |f_n| ≤ g):

∫(g + f) dμ = ∫liminf(g + fₙ) dμ
            ≤ liminf ∫(g + fₙ) dμ      (Fatou)
            = ∫g dμ + liminf ∫fₙ dμ.

So ∫f dμ ≤ liminf ∫fₙ dμ.

Now apply Fatou to g − f_n ≥ 0:

∫(g - f) dμ ≤ liminf ∫(g - fₙ) dμ
            = ∫g dμ - limsup ∫fₙ dμ.

So -∫f dμ ≤ -limsup ∫fₙ dμ, i.e., limsup ∫fₙ dμ ≤ ∫f dμ.

Combine: limsup ≤ ∫f ≤ liminf, forcing lim ∫f_n dμ = ∫f dμ. The stronger conclusion ∫|f_n − f| → 0 follows by applying the same argument to |f_n − f|, dominated by 2g.

JavaScript — visualizing the swap

// Verify ∫₀¹ sin(nx)/n dx → 0 numerically
function integralSinNXOverN(n, samples = 10000) {
  let sum = 0;
  const dx = 1 / samples;
  for (let i = 0; i < samples; i++) {
    const x = (i + 0.5) * dx;
    sum += Math.sin(n * x) / n;
  }
  return sum * dx;
}

for (const n of [1, 10, 100, 1000]) {
  console.log(`n=${n}:  ∫ = ${integralSinNXOverN(n).toExponential(3)}`);
}
// n=1:    ∫ = 4.597e-1
// n=10:   ∫ = 1.839e-2
// n=100:  ∫ = 1.000e-4
// n=1000: ∫ = 1.000e-6    ← shrinking like 1/n², matching analytical

// Counterexample: sliding rectangles
function slidingRectangleIntegral(n) {
  // fₙ(x) = n · 1_{(0, 1/n]}(x);  ∫fₙ = 1 always
  return 1.0;
}
// Pointwise: at any fixed x > 0, fₙ(x) = 0 for large n.
// But ∫fₙ = 1 for every n. DCT correctly refuses to apply
// (no integrable dominator).
console.log([1, 10, 100, 1000].map(slidingRectangleIntegral));
// [1, 1, 1, 1] — integrals stuck at 1 despite pointwise → 0

Comparison to other convergence theorems

Theorem	Hypothesis	Conclusion	Strength
Uniform convergence (Riemann)	sup|fn − f| → 0	∫fn → ∫f	Strongest — rarely satisfied
Monotone convergence (MCT)	0 ≤ fn ↑ f	∫fn → ∫f (possibly ∞)	Specific to monotone increasing
Fatou's lemma	fn ≥ 0	∫liminf fn ≤ liminf ∫fn	Inequality, no equality
Dominated convergence (DCT)	|fn| ≤ g, ∫g < ∞, fn → f a.e.	∫fn → ∫f and ∫|fn − f| → 0	Most flexible — signed functions
Vitali convergence	Uniform integrability + measure convergence	∫fn → ∫f	Weaker than DCT — no dominator needed
Bounded convergence	μ(X) < ∞, |fn| ≤ M constant	∫fn → ∫f	DCT special case — finite measure
Egoroff's theorem	μ(X) < ∞, fn → f a.e.	Uniform convergence on a set of large measure	Different conclusion — uniformity

Common pitfalls

• Forgetting the dominator. Pointwise convergence alone does NOT imply convergence of integrals — the sliding-rectangle counterexample shows this clearly. A fixed integrable g that caps every f_n is the essential ingredient.
• Using a dominator that depends on n. g must be the same function for every n. "|f_n| ≤ g_n" with varying g_n is useless — it's essentially saying |f_n| ≤ |f_n|.
• Confusing "dominator" with "upper bound at each x." The dominator must be integrable globally; an unbounded but pointwise-finite g may not be integrable.
• Treating "almost everywhere" as "everywhere." Pointwise a.e. convergence and a.e. domination are enough — exceptional sets of measure zero don't affect integrals. But the "everywhere" statements are subtly different.
• Believing DCT extends without modification to nets. DCT as stated handles sequences. For uncountable parameter families (parameter integrals), use continuous-parameter versions or restrict to suitable subsequences.
• Using DCT when uniform convergence is easier. If sup|f_n − f| → 0 and the measure space is finite, the uniform version of the convergence theorem is cleaner. DCT shines when uniform convergence fails but pointwise + dominator holds — Fourier series, parameter integrals, etc.

Applications

• Differentiation under the integral sign (Leibniz rule). ∂/∂t ∫f(x, t) dx = ∫∂f/∂t dx requires DCT applied to the difference quotients. If a single integrable g dominates the partial derivatives, the swap is justified.
• Fourier analysis. Convergence of Fourier partial sums in L² and L^p norms, term-by-term integration of Fourier series, the Riemann-Lebesgue lemma — all rest on DCT.
• Probability theory. Expectation continuity — if X_n → X a.s. and |X_n| ≤ Y with E[Y] < ∞, then E[X_n] → E[X]. Used in martingale convergence, central limit theorem proofs, characteristic-function arguments.
• Term-by-term integration of series. If Σ ∫|f_k| < ∞, then ∫Σf_k = Σ∫f_k. Direct corollary — apply DCT to partial sums with dominator Σ|f_k|.
• Continuity of integral transforms. Fourier, Laplace, Mellin transforms are continuous in their inputs in suitable L^p-topologies — DCT provides the limit-passing arguments.
• PDE — passing to the limit. Weak solutions to elliptic and parabolic PDE often require showing ∫u_n φ → ∫u φ for test functions φ; DCT is the workhorse.
• Numerical analysis — convergence of quadrature. Showing that piecewise-polynomial approximations converge in L¹ uses DCT with the polynomial approximation as the dominator.

History

Henri Lebesgue introduced the theorem in his 1908 paper as part of his programme to make integration robust enough to handle the convergence questions that Riemann integration could not address. Before Lebesgue, swapping limit and integral required uniform convergence — a very strong condition. After Lebesgue, the much weaker hypothesis "pointwise plus dominator" sufficed, unlocking dozens of arguments that had been blocked.

The theorem is often packaged with monotone convergence (Beppo Levi 1906, then Lebesgue) and Fatou's lemma (1906) as the three pillars of Lebesgue convergence theory. Probability (Kolmogorov 1933) and functional analysis (Banach 1932) were built on top.