1 Statement of Fatou’s Lemma

Fatou’s lemma provides a general inequality that relates a pointwise limiting behavior of functions to the corresponding behavior of their integrals. It is particularly useful when the only available structure is measurability and a one-sided bound (typically nonnegativity).

1.1 Nonnegative measurable functions

Let \((X,\mathcal{F},\mu)\) be a measure space and let \((f_n)_{n\ge 1}\) be a sequence of measurable functions taking values in \([0,\infty]\). The nonnegativity assumption ensures that the relevant integrals are well-defined as extended real numbers and supports monotonicity properties used in standard proofs.

1.2 Integral form (lim inf inequality)

Define the pointwise limit inferior by \[ \liminf_{n\to\infty} f_n(x)=\sup_{k\ge 1}\ \inf_{n\ge k} f_n(x). \] Fatou’s lemma states that \[ \int_X \liminf_{n\to\infty} f_n \, d\mu \ \le\ \liminf_{n\to\infty} \int_X f_n \, d\mu. \] Both sides may take the value \(+\infty\). If the right-hand side is finite, the inequality yields a meaningful finite upper bound for the left-hand side.

1.3 Probability-theoretic phrasing

On a probability space \((\Omega,\mathcal{G},\mathbb{P})\), Fatou’s lemma is often written as \[ \mathbb{E}\big[\liminf_{n\to\infty} X_n\big]\ \le\ \liminf_{n\to\infty}\mathbb{E}[X_n], \] for nonnegative random variables \(X_n\ge 0\). The same inequality interprets “expectation of a lower limit” as bounded above by “lower limit of expectations.”

1.4 Comparison with pointwise inequalities

A typical way to view the lemma is: even when one cannot interchange \(\liminf\) and integration, the inequality guarantees that integration does not overshoot the limiting lower envelope. In contrast to pointwise statements like \(\liminf f_n(x)\le \limsup f_n(x)\), Fatou connects these pointwise operations to global integral quantities.

2 Measurability and Preconditions

The lemma’s usefulness depends on verifying that the objects involved are measurable and that the limiting expression is meaningful.

2.1 Measurable functions and extended real values

Measurability ensures that \(\int f_n\,d\mu\) is defined and that operations like \(\inf_{n\ge k} f_n\) and \(\sup_{k\ge 1}\) preserve measurability. Using extended real values allows inclusion of cases where integrals diverge to \(+\infty\).

2.2 Why nonnegativity matters

Nonnegativity prevents sign-cancellation from destroying the inequality. For signed functions, the integral of the limit inferior need not be bounded by the limit inferior of the integrals; additional assumptions are required to control negative parts. The nonnegative hypothesis is the minimal condition under which Fatou’s lemma holds in its basic form.

2.3 Handling infinite integrals

If \(\int f_n\,d\mu\) diverges along a subsequence, the right-hand side becomes \(+\infty\) and the inequality is still valid. If \(\int \liminf f_n\,d\mu=+\infty\), then the inequality holds trivially unless the right-hand side is finite; in practice, applications often use it to show that finiteness of the limit inferior of integrals forces finiteness of \(\int \liminf f_n\).

3 Proof Strategies

Several proofs exist; the most common rely on truncation or on monotonicity tools such as the monotone convergence theorem.

3.1 Proof via truncation (using positive parts)

A standard approach truncates the functions from below. For each \(m\in\mathbb{N}\), define \[ f_n^{(m)} := \min(f_n,m). \] Then \(0\le f_n^{(m)}\le m\), and \(\int f_n^{(m)}\,d\mu\) is finite for each fixed \(m\). One shows that \[ \liminf_{n\to\infty} f_n^{(m)} = \min\!\big(\liminf_{n\to\infty} f_n,\, m\big). \] Applying a monotone convergence argument in \(m\) (as \(m\uparrow\infty\)) yields the Fatou inequality for the original \(f_n\). This truncation step isolates the key lower-limit behavior while avoiding complications from infinite values.

3.2 Proof using monotone convergence theorem

Let \[ g_k(x):=\inf_{n\ge k} f_n(x). \] Then \(g_k\) increases with \(k\) and \[ \lim_{k\to\infty} g_k(x)=\liminf_{n\to\infty} f_n(x). \] Since \(g_k\ge 0\), the monotone convergence theorem implies \[ \int \liminf_{n\to\infty} f_n\, d\mu=\int \lim_{k\to\infty} g_k\,d\mu = \lim_{k\to\infty}\int g_k\,d\mu. \] Next, for each \(k\), \(g_k \le f_n\) for all \(n\ge k\), hence \(\int g_k\,d\mu \le \inf_{n\ge k}\int f_n\,d\mu\). Taking \(\lim_{k\to\infty}\) gives the desired inequality: \[ \int \liminf f_n\,d\mu \le \liminf \int f_n\,d\mu. \]

3.3 Proof using lim inf characterization

Using the characterization \(\liminf f_n = \sup_k \inf_{n\ge k} f_n\), one can build an increasing sequence from the infima tail functions and then integrate using monotonicity. The core idea is that “lim inf” is a supremum of measurable functions that come from tails; integration then respects the monotone structure.

3.4 Alternative arguments and measure-space viewpoints

Other perspectives include:

  • Order-theoretic view: Fatou’s lemma reflects that the integral is a monotone functional compatible with suprema of increasing sequences.
  • Semicontinuity view: The inequality expresses lower semicontinuity of an integral functional under pointwise lim inf limits.
  • Capacity-style comparisons: In some settings, the proof is adapted to semi-finite or \(\sigma\)-finite measures by reducing to sets of finite measure and applying the core lemma there.

Fatou’s lemma interacts with other convergence principles and admits several companion inequalities.

4.1 Reverse Fatou’s lemma (upper bounds)

A common “reverse” statement typically requires additional control beyond nonnegativity—often boundedness of integrals from above, uniform integrability, or assumptions on the lim sup. Without extra hypotheses, there is no general inequality of the form \[ \int \limsup f_n\,d\mu \ge \limsup \int f_n\,d\mu \] for nonnegative \(f_n\). Reverse-type results are obtained under supplementary conditions that prevent mass from escaping to infinity.

4.2 Fatou’s lemma in terms of lim sup

Applying Fatou to complements or to transformed sequences can yield inequalities involving \(\limsup\). For instance, when one has negative functions or subtraction structures, reformulating them as differences of nonnegative terms allows one to express bounds in terms of lim sup, but typically with additional assumptions to ensure integrability of parts.

4.3 For sequences of random variables

When \(X_n\ge 0\), the lemma immediately yields \[ \mathbb{E}\big[\liminf X_n\big]\le \liminf \mathbb{E}[X_n]. \] This is frequently used in probability to compare almost-sure limits with expectation bounds. In applications to convergence in distribution, it plays a role similar to lower bounds on expectations even when full convergence of expectations fails.

4.4 Fatou-type inequalities with parameters

One sometimes considers parameterized families \(f_{n,\theta}\) and seeks uniform bounds of the form \[ \int \liminf_{n} f_{n,\theta}\,d\mu \le \liminf_{n} \int f_{n,\theta}\,d\mu, \] with \(\theta\) acting as an external index. Under measurability assumptions in \(\theta\), the lemma can be applied pointwise in \(\theta\), or combined with further arguments to exchange limits and integrals in multi-parameter settings.

5 Applications in Analysis and Probability

Fatou’s lemma is a tool for producing inequalities when limits and integrals cannot be interchanged directly.

5.1 Establishing lower semicontinuity of integral functionals

Consider an integral functional \(I(f)=\int f\,d\mu\) on nonnegative measurable functions. If \(f_n\to f\) pointwise in the sense that \(f\le \liminf f_n\), then Fatou yields \[ I(f)\le \liminf_{n\to\infty} I(f_n). \] This is one of the standard ways to verify lower semicontinuity properties used in variational calculus.

5.2 Tightness and convergence of expectations (inequality form)

In probability, one often has random variables \(X_n\ge 0\) converging almost surely to \(X\). Fatou implies \[ \mathbb{E}[X]\le \liminf_{n\to\infty}\mathbb{E}[X_n]. \] Thus, if \(\sup_n\mathbb{E}[X_n]\) is controlled, Fatou provides constraints on the expectation of the limiting random variable. While it does not guarantee equality, it still yields a robust one-sided statement.

5.3 Existence and bounds in variational arguments

Variational methods often generate minimizing sequences \(f_n\) with energies \(\int \Phi(f_n)\,d\mu\). If one can show \(\Phi(f)\le \liminf \Phi(f_n)\) pointwise and \(\Phi(f_n)\ge 0\), then Fatou gives \[ \int \Phi(f)\,d\mu \le \liminf_{n\to\infty} \int \Phi(f_n)\,d\mu, \] which is instrumental in proving that limit candidates have energy no worse than the infimum.

5.4 Using Fatou to pass to the limit in inequalities

Suppose inequalities of the form \(\int f_n\,d\mu \ge a_n\) are known and \(f_n\) has a lim inf structure. Fatou can be combined with bounds on \(a_n\) to derive \[ \int \liminf f_n\,d\mu \le \liminf \int f_n\,d\mu, \] and then infer relationships between the limiting integral and the limiting lower bounds. This is particularly useful when only weak information on pointwise convergence is available.

6 Connections to Convergence Theorems

Fatou’s lemma is closely linked to the monotone and dominated convergence theorems, each of which supplies additional hypotheses to upgrade inequalities into equalities.

6.1 Relation to monotone convergence theorem

If \(f_n\uparrow f\) pointwise with \(f_n\ge 0\), then \(\liminf f_n=f\). Fatou’s lemma gives \[ \int f\,d\mu \le \liminf \int f_n\,d\mu, \] while monotonicity of the integrals provides the reverse inequality, yielding equality. Thus monotone convergence can be viewed as a case where Fatou’s inequality becomes sharp.

6.2 Relation to dominated convergence theorem

Dominated convergence provides equality under an integrable dominating function. In such cases, \(f_n\to f\) in a way strong enough that \(\liminf f_n\) and \(\limsup f_n\) both equal \(f\) almost everywhere. Fatou’s lemma yields one side of the desired equality, and applying it to a suitable transformation supplies the other side, with domination ensuring integrability of the relevant terms.

6.3 Comparison with uniform integrability (expectation control)

Uniform integrability refines convergence of expectations beyond almost-sure convergence. Fatou supplies a universal lower bound for expectations of lim infs, but uniform integrability is what typically upgrades that to \[ \mathbb{E}[X_n]\to \mathbb{E}[X] \] when \(X_n\to X\) in probability and other standard assumptions hold. Conceptually, uniform integrability prevents the “loss of mass” phenomenon that would otherwise make Fatou strict.

6.4 When Fatou is sharp or insufficient

Fatou is often strict when mass concentrates along subsequences and disappears in the limit inferior. It becomes insufficient for obtaining upper bounds or equality without extra conditions. In practical analysis, this guides the choice: one may need dominated convergence or uniform integrability to strengthen Fatou’s inequality.

7 Extensions and Generalizations

Beyond its classical statement for nonnegative measurable functions, Fatou’s lemma has broader variants adapted to different algebraic and measure-theoretic settings.

7.1 Complex- or signed-function adaptations where applicable

For signed functions \(f_n\), a direct application is not valid. However, one can sometimes apply Fatou to nonnegative parts, such as the positive part \(f_n^+\) and negative part \(f_n^-\), under integrability assumptions that control either part. For complex-valued functions, one typically applies the lemma to \(f_n\) or to nonnegative quantities derived from real components, again requiring suitable assumptions to relate the absolute-value bounds to the desired conclusion.

7.2 Vector-valued versions

In spaces where functions take values in \(\mathbb{R}^m\) or other ordered structures, one may apply scalar Fatou to norms, coordinates, or positive functionals. Care is needed because order properties used in the scalar lemma may not transfer directly to vector spaces without additional structure (for example, componentwise ordering).

7.3 Applications to product measures and iterated integrals

When dealing with product spaces or iterated integration, Fatou’s lemma can be applied to one variable at a time under measurability and integrability conditions. This supports inequalities for marginal limits and helps justify passing to limits in iterated integrals, though full interchange of limits may still require dominated convergence or Fubini-type theorems with appropriate hypotheses.

8 Examples and Worked Computations

Examples illustrate how the inequality operates and when it becomes strict.

8.1 Simple deterministic examples

Let \(X=[0,1]\) with Lebesgue measure. Define \(f_n(x)=\mathbf{1}_{[0,1/n]}(x)\). Then \(f_n(x)\to 0\) for all \(x>0\), while at \(x=0\) the sequence equals \(1\) for all \(n\). The limit inferior satisfies \[ \liminf_{n\to\infty} f_n(x)=0 \quad \text{for all } x\in(0,1], \] and equals \(1\) at \(x=0\). Since a single point has measure zero, \(\int \liminf f_n\,dx=0\). Meanwhile \(\int f_n\,dx = 1/n\), so \(\liminf \int f_n\,dx=0\). Here equality holds, but the example highlights that pointwise behavior at null sets does not affect integrals.

8.2 Indicator-function examples

Using indicator functions provides an intuitive picture of “mass moving around.” Consider \(f_n=\mathbf{1}_{A_n}\) where \((A_n)\) is a sequence of measurable sets with \(\mu(A_n)\) constant but the sets shift so that points belong to only finitely many \(A_n\). If \(\liminf \mathbf{1}_{A_n}=\mathbf{1}_{\liminf A_n}\) captures the set of points that are in all but finitely many \(A_n\), then Fatou compares the measure of \(\liminf A_n\) with the lim inf of \(\mu(A_n)\). This makes it clear that the lim inf set measures the persistent portion of the sequence.

8.3 Random-variable examples with lim inf behavior

Let \(X_n(\omega)\ge 0\) and suppose \(X_n\to X\) almost surely but expectations are not known to converge. Fatou ensures \[ \mathbb{E}[X]\le \liminf_{n\to\infty}\mathbb{E}[X_n]. \] For instance, if \(X_n\) are nonnegative and \(\mathbb{E}[X_n]\) remains bounded below by a constant \(c\), then the limiting random variable must also have expectation at least \(c\). This can be used to rule out candidate limits in stochastic models.

8.4 Cases illustrating strict inequality vs equality

Strictness occurs when the lim inf loses mass relative to the integrals. One way to see this is to construct \(f_n\ge 0\) such that each \(f_n\) has substantial integral but the pointwise lim inf is small almost everywhere. For shifted spikes, the mass “moves” so that any fixed point sees values only rarely. Then \(\liminf f_n(x)\) can be zero almost everywhere, making \[ \int \liminf f_n\,d\mu = 0 \] while \(\liminf \int f_n\,d\mu\) is positive. Equality, by contrast, holds in settings where the mass persists in the lim inf, such as monotone increasing sequences or dominated convergence situations where \(f_n\to f\) and the integrals align.