Monotone and Dominated Convergence

Part 12, Chapter 12: Measure Theory and the Lebesgue Integral

Learning objectives

State and apply the Monotone Convergence Theorem (MCT)
State and apply the Dominated Convergence Theorem (DCT)
State Fatou's Lemma and exhibit a sequence where the inequality is strict
Use the theorems to interchange limits and integrals in concrete computations

The Riemann integral has a single deep failing: you usually cannot interchange limit and integral. If $f_n \to f$ ntof pointwise, it is shockingly difficult to conclude $\int f_n \to \int f$ ntointf in the Riemann theory, you need uniform convergence, which is a strong and rare condition. The Lebesgue integral fixes this. The three convergence theorems of this section, Monotone, Fatou, Dominated, give clean, broadly applicable, easy-to-check conditions under which $\lim$ and $\int$ commute. Every limit interchange you will ever do in probability, harmonic analysis, or differential equations cites one of these three results.

The convergence theorems are statements about general measure spaces and do not have a single specific widget in lang-core. We motivate them with three explicit sequences of step functions, the canonical examples in every real-analysis textbook.

Monotone Convergence Theorem (MCT)

Statement. Let $\{f_n\}$ n be a sequence of measurable functions on $(X, \mathcal{A}, \mu)$ with $0 \leq f_n \leq f_{n+1}$ pointwise (a.e.) and $f_n \to f$ ntof pointwise (a.e.). Then

$\int f\, d\mu = \lim_{n \to \infty} \int f_n\, d\mu.$ ,dmu.

Mnemonic: if the sequence climbs monotonically, the integral climbs with it, nothing escapes. Example: take $f_n = \chi_{[0, n]}$ on $\mathbb{R}$ . Each $f_n \leq f_{n+1}$ , the pointwise limit is $f = \chi_{[0, \infty)}$ [0,infty), and

$\int f_n\, d\lambda = n \nearrow \infty = \int f\, d\lambda,$ n,dlambda=nnearrowinfty=intf,dlambda,

so MCT gives the expected (if infinite) answer.

Fatou's Lemma

Statement. If $\{f_n\}$ n are non-negative measurable functions, then

$\int \liminf_{n \to \infty} f_n\, d\mu \;\leq\; \liminf_{n \to \infty} \int f_n\, d\mu.$ n,dmu.

This inequality can be strict. The classic example is $f_n = n\, \chi_{[0, 1/n]}$ : for every $x > 0$ eventually $x > 1/n$ , so $f_n(x) \to 0$ n(x)to0. The liminf of the functions is $0$ , so its integral is $0$ . But $\int f_n\, d\lambda = n \cdot (1/n) = 1$ n,dlambda=ncdot(1/n)=1 for every $n$ , so the liminf of the integrals is $1$ . Fatou says $0 \leq 1$ , correct, and the inequality is strict.

Intuition: mass can "leak out to infinity" or "concentrate at a point" in such a way that the integrals all stay positive while the pointwise limit is zero. Fatou is a one-way safeguard: the integral of the limit is at most the limit of the integrals, never more.

Dominated Convergence Theorem (DCT)

Statement. Let $\{f_n\}$ n be measurable functions with $f_n \to f$ ntof pointwise (a.e.). Suppose there exists a non-negative integrable function $g$ with $|f_n(x)| \leq g(x)$ n(x)∣leqg(x) for almost every $x$ and every $n$ . Then $f$ is integrable and

$\lim_{n \to \infty} \int f_n\, d\mu = \int f\, d\mu, \qquad \lim_{n \to \infty} \int |f_n - f|\, d\mu = 0.$ ,dmu=intf,dmu,qquadlimntoinftyint∣fn−f∣,dmu=0.

The dominating function $g$ is the entire point: it caps the size of every $f_n$ n uniformly and prevents mass from escaping to infinity or piling up at a single point. With $g$ in place, swap-of-limit-and-integral is legal. Without $g$ , the example $f_n = n\, \chi_{[0, 1/n]}$ shows the interchange can fail spectacularly: pointwise limit is $0$ , integrals all equal $1$ .

Three sequences, three behaviours

$f_n = \chi_{[0, n]}$ : monotone increasing, no dominator, integral diverges to $\infty$ . MCT applies; gives $\int f = \infty$ .
$f_n = n\, \chi_{[0, 1/n]}$ : not monotone, no dominator (the only candidate would be the integrable $g(x) = 1/x$ near $0$ , which is not integrable), Fatou gives $0 \leq 1$ strictly, DCT does not apply.
$f_n(x) = \frac{n x}{1 + n^2 x^2}$ n(x)=fracnx1+n2x2 on $[0, 1]$ : pointwise limit is $0$ , dominated by $g(x) = 1/2$ (because $1 + n^2 x^2 \geq 2 n x$ by AM-GM), DCT applies and gives $\lim \int f_n = 0$ n=0.

Where this shows up

Quantum mechanics: Time-evolved wave functions are written as integrals of energy-eigenstate components against the spectral measure. Computing $\langle \psi(t), A \psi(t) \rangle$ for an observable $A$ as $t \to \infty$ uses DCT pointwise on $t$ , dominated by $|\psi|^2$ which is in $L^1$ for normalised states.
Probability theory: Almost-sure convergence of random variables combined with a uniform integrability hypothesis gives $E[X_n] \to E[X]$ , DCT in the probabilistic dialect. The weak and strong laws of large numbers and central limit theorem proofs all rely on Fatou and DCT at key steps.
Heat equation and PDE: Solutions of $\partial_t u = \Delta u$ tu=Deltau are represented as convolutions with the heat kernel. Showing that the solution converges to the initial data as $t \to 0^+$ uses DCT with the heat kernel itself as the dominator.
Machine learning: Stochastic gradient methods compute expectations by averaging samples; convergence of the empirical risk to the true risk under iid sampling is a DCT/Fatou argument. Even more directly, the chain rule for backpropagation through an integral (a "score-function estimator") uses DCT to swap differentiation and expectation.

Pause and think: Why does MCT not apply to the sequence $f_n = n\, \chi_{[0, 1/n]}$ ? Walk through the hypotheses one by one. (Hint: is the sequence monotone increasing?) Why does DCT not apply? (Hint: is there an integrable function that dominates every $f_n$ n?)

Try it

Predict first: compute $\lim_{n \to \infty} \int_0^1 x^n\, dx$ ,dx using DCT with the dominator $g(x) = 1$ . (Hint: pointwise $x^n \to 0$ for $x \in [0, 1)$ and stays $1$ at $x = 1$ .)
Apply MCT (which needs non-negative monotone terms) to compute $\int_0^1 \sum_{n=0}^\infty x^n / n!\, dx = \int_0^1 e^x\, dx$ and confirm by integrating term-by-term.
True or false: if $f_n \to f$ ntof pointwise on $[0, 1]$ , then $\int f_n \to \int f$ ntointf. (Hint: counterexample with $f_n = n \chi_{[0, 1/n]}$ on $[0, 1]$ .)
Fatou check: exhibit a sequence where the Fatou inequality is an equality. (Hint: any monotone increasing sequence, by MCT.)
DCT in action: evaluate $\lim_{n \to \infty} \int_0^\infty \frac{\sin(x/n)}{x(1+x^2)}\, dx$ . (Hint: pointwise the integrand $\to 0$ ; using $|\sin(t)/t|\leq 1$ , dominator is $g(x) = 1/(1+x^2)$ , verify it is integrable on $[0,\infty)$ ; conclude limit is $0$ .)

A trap to watch for

A subtle and very common error is to apply DCT without checking that a dominator exists, people quote the theorem and then forget the $|f_n| \leq g$ n∣leqg clause. Worse, they sometimes try to use $g = \sup_n |f_n|$ as the dominator, which is fine only if that supremum is itself integrable. The sequence $f_n = n\, \chi_{[0, 1/n]}$ has $\sup_n |f_n|(x) = \infty$ at $x = 0$ and behaves like $1/x$ near zero, which is not integrable. There is no valid dominator and the theorem genuinely does not apply. Always write down $g$ explicitly and check $\int g\, d\mu < \infty$ before quoting DCT. If you cannot find such a $g$ , look for monotonicity (use MCT) or fall back on Fatou's one-sided inequality.

What you now know

You can state the Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem, recognise which one applies in concrete situations, and use them to compute $\lim_n \int f_n$ in cases where the Riemann theory is silent. You have also seen the canonical "escape to infinity" counterexample ( $f_n = n\,\chi_{[0, 1/n]}$ ) that motivates the dominator condition. This closes the introduction to measure theory and Lebesgue integration; the further development, $L^p$ spaces, Fubini-Tonelli, Radon-Nikodym, all builds on these three theorems.

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 12.
Royden, H. L., Fitzpatrick, P. M. (2010). Real Analysis (4th ed.). Pearson, ch. 4.
Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications (2nd ed.). Wiley, ch. 2.
Rudin, W. (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill, ch. 1.
Stein, E. M., Shakarchi, R. (2005). Real Analysis. Princeton UP, ch. 2-3.