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Monotone convergence theorem

The monotone convergence theorem is a cornerstone result in measure theory, asserting that if a sequence of non-negative measurable functions \{f_n\} on a measure space (X, \mathcal{A}, \mu) increases pointwise almost everywhere to a limit function f, then the integral of the limit equals the limit of the integrals: \int_X f \, d\mu = \lim_{n \to \infty} \int_X f_n \, d\mu.^[1] This theorem, first proved by Henri Lebesgue in his 1902 dissertation and later generalized by Beppo Levi in 1906,^[2] extends the basic property of Riemann integrals to the more general Lebesgue framework, enabling the rigorous handling of limits under integration for monotone sequences.^[3] It applies specifically to non-decreasing sequences of functions taking values in [0, \infty], requiring measurability and pointwise convergence almost everywhere, and serves as a prerequisite for more advanced results like the dominated convergence theorem.^[3] The theorem's proof typically relies on Fatou's lemma and the monotonicity of the Lebesgue integral, highlighting its role in establishing continuity of the integral operator with respect to monotone limits.^[1] In applications, it justifies interchanging limits and integrals in probability theory, partial differential equations, and Fourier analysis, where constructing integrals via limits of simple functions is common.^[3]

Sequences and Series in Real Numbers

Monotone Sequence Convergence

A monotone sequence of real numbers is one that is either non-decreasing or non-increasing. Specifically, a sequence \{a_n\}_{n=1}^\infty is monotone increasing if a_n \leq a_{n+1} for all n \in \mathbb{N}, and strictly increasing if the inequality is strict; it is monotone decreasing if a_n \geq a_{n+1} for all n \in \mathbb{N}, and strictly decreasing if strict.^[4] The monotone convergence theorem for sequences states that every bounded monotone sequence of real numbers converges. More precisely, if \{a_n\} is monotone increasing and bounded above, then it converges to its least upper bound, or supremum:

\lim_{n \to \infty} a_n = \sup \{a_n : n \in \mathbb{N}\}.

Similarly, if \{a_n\} is monotone decreasing and bounded below, it converges to its greatest lower bound, or infimum. Conversely, a monotone sequence converges if and only if it is bounded.^[5]^[6] To prove this, consider the case of a monotone increasing sequence \{a_n\} that is bounded above. Let S = \{a_n : n \in \mathbb{N}\}, and let L = \sup S, which exists by the completeness axiom of the real numbers (the least upper bound property). For any \epsilon > 0, L - \epsilon is not an upper bound for S, so there exists some N \in \mathbb{N} such that a_N > L - \epsilon. Since the sequence is increasing, a_n \geq a_N > L - \epsilon for all n \geq N. Also, a_n \leq L for all n, so L - \epsilon < a_n \leq L for all n \geq N, proving that \lim_{n \to \infty} a_n = L. The decreasing case follows analogously by considering -a_n. The "only if" direction holds because every convergent sequence is bounded. This proof relies on the Dedekind completeness of the reals, ensuring every nonempty subset bounded above has a least upper bound.^[4]^[7]^[8] This foundational result in real analysis relies on the completeness of the real numbers, developed in the 19th century.

Monotone Series Convergence

A monotone series is defined as an infinite series \sum_{n=1}^\infty a_n where each term a_n \geq 0 for all n \in \mathbb{N}, ensuring that the sequence of partial sums S_n = \sum_{k=1}^n a_k is non-decreasing, or increasing if all a_n > 0.^[9] The fundamental theorem for such series states that \sum_{n=1}^\infty a_n converges if and only if the partial sums \{S_n\} are bounded above by some finite M > 0. If bounded, the series converges to \lim_{n \to \infty} S_n = \sup \{ S_n : n \in \mathbb{N} \}; otherwise, the partial sums diverge to \infty, and the series diverges.^[9] The proof follows directly from the monotone convergence theorem for sequences applied to the partial sums. Since \{S_n\} is monotone increasing (as S_{n+1} = S_n + a_{n+1} \geq S_n) and bounded above if and only if the series converges, the limit exists and is finite precisely when bounded.^[9] This result applies exclusively to series with non-negative terms and differs from absolute convergence, which requires the series of absolute values \sum |a_n| to converge for general real or complex terms, guaranteeing unconditional convergence but not vice versa in the non-negative case.^[9]

Examples of Monotone Sums

The harmonic series \sum_{n=1}^\infty \frac{1}{n} provides a classic example of a monotone divergent series. Its partial sums H_n = \sum_{k=1}^n \frac{1}{k} form a monotonically increasing sequence because each term is positive, yet H_n is unbounded above, as H_n > \ln n + \gamma where \gamma \approx 0.577 is the Euler-Mascheroni constant, implying divergence to infinity.^[10]^[9] The Taylor series for the base of the natural logarithm, e = \sum_{k=0}^\infty \frac{1}{k!}, exemplifies monotone convergence of a series with positive terms. The partial sums S_n = \sum_{k=0}^n \frac{1}{k!} are monotonically increasing and bounded above by 3, since the tail satisfies e - S_n = \sum_{k=n+1}^\infty \frac{1}{k!} < \frac{1}{n!} \sum_{j=0}^\infty \frac{1}{(n+1)^j} = \frac{1}{n! (n)} for n \geq 1, implying S_n < e < S_n + \frac{1}{n!}.^[11] Thus, by the monotone convergence theorem for series, S_n converges to e \approx 2.71828.^[11]

Measure-Theoretic Generalization

Beppo Levi's Lemma for Integrals

Beppo Levi's lemma, named after the Italian mathematician Beppo Levi (1875–1961), emerged in the early 20th century as a cornerstone of the developing theory of Lebesgue integration, building directly on Henri Lebesgue's foundational work from 1902.^[12] Levi's contributions, particularly in his 1906 publications, addressed key aspects of integration for non-negative functions and series, providing rigorous justifications that complemented and extended Lebesgue's ideas.^[13] This lemma represents the measure-theoretic generalization of the monotone convergence theorem for real sequences, adapting the discrete concept to integrals over abstract measure spaces. In the context of Lebesgue integration, consider a measure space (X, \mathcal{M}, \mu). A simple function is a finite sum s = \sum_{k=1}^m c_k \chi_{E_k}, where c_k \geq 0, the E_k are disjoint measurable sets in \mathcal{M}, and \chi_{E_k} is the characteristic function of E_k. The integral of such a simple function is \int s \, d\mu = \sum_{k=1}^m c_k \mu(E_k). For a non-negative measurable function f: X \to [0, \infty], the Lebesgue integral is defined as

\int_X f \, d\mu = \sup \left\{ \int_X s \, d\mu : s \text{ simple}, \, 0 \leq s \leq f \right\},

which may equal \infty. This definition extends the integral from simple functions to all non-negative measurable functions via approximation from below. Beppo Levi's lemma asserts that if \{f_n\}_{n=1}^\infty is a sequence of non-negative measurable functions on X such that f_n \uparrow f pointwise almost everywhere (meaning $0 \leq f_1 \leq f_2 \leq \cdots and \lim_{n \to \infty} f_n(x) = f(x) for \mu-almost all x \in X, where f is the measurable function equal almost everywhere to this pointwise limit), then

\int_X f \, d\mu = \lim_{n \to \infty} \int_X f_n \, d\mu.

The integrals take values in [0, \infty], and the limit exists (possibly infinite) because the sequence \{\int_X f_n \, d\mu\}_{n=1}^\infty is non-decreasing by the monotonicity of the Lebesgue integral for non-negative functions. This equality follows from the definition of the integral, as the partial approximations align under the pointwise limit almost everywhere:

\int_X f \, d\mu = \sup \left\{ \int_X s \, d\mu : s \text{ simple}, \, 0 \leq s \leq f \right\} = \lim_{n \to \infty} \int_X f_n \, d\mu,

ensuring the interchange of limit and integral under monotonicity.^[14]

Monotonicity of the Lebesgue Integral

The monotonicity property of the Lebesgue integral states that if $0 \leq g \leq f are non-negative measurable functions on a measure space (X, \mathcal{A}, \mu), then \int_X g \, d\mu \leq \int_X f \, d\mu.^[15] This property is fundamental to the theory of integration and serves as a prerequisite for results like Beppo Levi's lemma.^[16] To prove this, first consider the case where f and g are simple functions. Express f = \sum_{i=1}^n a_i \chi_{E_i}, where a_i \geq 0, the E_i are disjoint measurable sets partitioning the support of f, and \chi_{E_i} is the characteristic function of E_i. The Lebesgue integral is then \int_X f \, d\mu = \sum_{i=1}^n a_i \mu(E_i).^[17] Since $0 \leq g \leq f, on each E_i, g \leq a_i, so \int_{E_i} g \, d\mu \leq a_i \mu(E_i). Summing over i gives \int_X g \, d\mu \leq \sum_{i=1}^n a_i \mu(E_i) = \int_X f \, d\mu.^[16] For general non-negative measurable functions f and g with g \leq f, the result follows directly from the definition: any simple s with $0 \leq s \leq g also satisfies s \leq f, so

\int_X g \, d\mu = \sup\left\{ \int_X s \, d\mu : s \text{ simple}, 0 \leq s \leq g \right\} \leq \sup\left\{ \int_X t \, d\mu : t \text{ simple}, 0 \leq t \leq f \right\} = \int_X f \, d\mu.

A direct consequence is that if \{f_n\} is a sequence of non-negative measurable functions with f_n \uparrow f pointwise almost everywhere, then \int_X f_n \, d\mu \uparrow \int_X f \, d\mu. This follows by repeated application of the monotonicity property to the differences f_{n+1} - f_n \geq 0.^[17] Unlike the Riemann integral, which may fail for functions with many discontinuities due to reliance on interval partitions, the Lebesgue integral's monotonicity holds robustly through measure-theoretic approximation, accommodating arbitrary non-negative measurable functions.^[18]

Proof of Beppo Levi's Lemma

Beppo Levi's lemma states that if \{f_n\}_{n=1}^\infty is an increasing sequence of nonnegative measurable functions on a measure space (X, \mathcal{A}, \mu), then f = \lim_{n \to \infty} f_n (defined almost everywhere) is measurable and

\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu,

where the integrals may be infinite.^[19] The proof proceeds in two main steps, beginning with the case where the f_n are simple functions. In this case, each f_n = \sum_{i=1}^{m_n} a_{n,i} \chi_{E_{n,i}} with a_{n,i} \geq 0 and E_{n,i} \in \mathcal{A}. Since the sequence is increasing, f(x) = \sup_n f_n(x) for \mu-almost every x \in X, and the integral \int_X f_n \, d\mu = \sum_{i=1}^{m_n} a_{n,i} \mu(E_{n,i}) increases with n by monotonicity of the measure. The pointwise limit f is a nonnegative measurable function, and since each f_n is simple with f_n \leq f, \lim \int f_n \leq \int f; the reverse holds by the sup definition, as simple functions below f are eventually approximated by the f_n.^[1] For the general case, approximate each nonnegative measurable f_n by an increasing sequence of simple functions s_{n,k} \uparrow f_n as k \to \infty, such that \int_X s_{n,k} \, d\mu \uparrow \int_X f_n \, d\mu by the definition of the Lebesgue integral for nonnegative functions. The double-indexed sequence \{s_{n,k}\} then satisfies s_{n,k} \uparrow f as n,k \to \infty almost everywhere. By monotonicity of the integral,

\lim_{n \to \infty} \int_X f_n \, d\mu = \lim_{n \to \infty} \lim_{k \to \infty} \int_X s_{n,k} \, d\mu = \lim_{k \to \infty} \lim_{n \to \infty} \int_X s_{n,k} \, d\mu = \lim_{k \to \infty} \int_X \lim_{n \to \infty} s_{n,k} \, d\mu = \int_X f \, d\mu,

where the iterated limits commute due to the joint monotonicity in both indices, as the double sequence increases to f.^[20] Monotonicity of the Lebesgue integral ensures that \int_X f_n \, d\mu \leq \int_X f \, d\mu for each n, so \lim_{n \to \infty} \int_X f_n \, d\mu \leq \int_X f \, d\mu. The reverse inequality follows from the approximation: for any simple function \phi with $0 \leq \phi \leq f, define A_n = \{x : f_n(x) \geq t \phi(x)\} for $0 < t < 1; then A_n \uparrow X almost everywhere, and \int_X f_n \, d\mu \geq t \int_{A_n} \phi \, d\mu. Taking n \to \infty yields \lim_{n \to \infty} \int_X f_n \, d\mu \geq t \int_X \phi \, d\mu by continuity of the measure from below. Letting t \to 1^- gives \lim_{n \to \infty} \int_X f_n \, d\mu \geq \int_X \phi \, d\mu. Supremum over all such \phi yields \lim_{n \to \infty} \int_X f_n \, d\mu \geq \int_X f \, d\mu, establishing equality by squeezing.^[19] If \lim_{n \to \infty} \int_X f_n \, d\mu = \infty, then \int_X f \, d\mu = \infty, as the reverse inequality always holds. This completes the proof, with the monotonicity of the integral serving as the key tool throughout.^[21]

Advanced Proof Techniques

Intermediate Lemmas for Convergence

In measure theory, several intermediate lemmas establish key properties of measures that facilitate proofs of convergence theorems, particularly by linking set measures to integrals via characteristic functions. These lemmas exploit the structure of increasing sequences of sets and the monotonicity of the underlying measure. A fundamental result is the continuity of measures from below, which states that if \{E_n\}_{n=1}^\infty is an increasing sequence of measurable sets with E = \bigcup_{n=1}^\infty E_n, then \mu(E) = \lim_{n \to \infty} \mu(E_n) for a measure \mu on a measurable space. To prove this, consider the characteristic functions \chi_{E_n}, which form an increasing sequence converging pointwise to \chi_E. Since the integral of a characteristic function equals the measure of the corresponding set, applying Beppo Levi's lemma to the nonnegative measurable functions \chi_{E_n} \uparrow \chi_E yields \int \chi_E \, d\mu = \lim_{n \to \infty} \int \chi_{E_n} \, d\mu, or equivalently, \mu(E) = \lim_{n \to \infty} \mu(E_n). This lemma can also be established using properties of outer measures, but the integral approach highlights its connection to functional convergence. Building on this, another lemma extends finite additivity to countable unions of disjoint measurable sets. Specifically, if \{E_n\}_{n=1}^\infty is a sequence of pairwise disjoint measurable sets with E = \bigcup_{n=1}^\infty E_n, then the measure satisfies countable additivity: \mu(E) = \sum_{n=1}^\infty \mu(E_n). The proof proceeds by defining partial unions F_k = \bigcup_{n=1}^k E_n, which form an increasing sequence converging to E and satisfy \mu(F_k) = \sum_{n=1}^k \mu(E_n) by finite additivity; continuity from below then implies \mu(E) = \lim_{k \to \infty} \mu(F_k) = \sum_{n=1}^\infty \mu(E_n). These lemmas serve as bridges between finite additivity—assumed in the definition of a premeasure—and the \sigma-additivity required for a full measure, enabling rigorous extensions to infinite processes in convergence arguments.^[22]

Proof Using Fatou's Lemma

Fatou's lemma provides a fundamental inequality for interchanging limits and integrals of non-negative functions. Specifically, if \{f_n\} is a sequence of non-negative measurable functions on a measure space (X, \mathcal{M}, \mu), then

\int_X \liminf_{n \to \infty} f_n \, d\mu \leq \liminf_{n \to \infty} \int_X f_n \, d\mu.

This result, originally established by Pierre Fatou in his 1906 work on trigonometric series, forms the basis for many convergence theorems in measure theory.^[23] To prove the monotone convergence theorem using Fatou's lemma, assume \{f_n\} is a sequence of non-negative measurable functions such that $0 \leq f_1 \leq f_2 \leq \cdots and f_n \uparrow f pointwise, where f is also non-negative and measurable. Since the sequence is monotonically increasing, \liminf_{n \to \infty} f_n = \lim_{n \to \infty} f_n = f. Applying Fatou's lemma yields

\int_X f \, d\mu \leq \liminf_{n \to \infty} \int_X f_n \, d\mu.

The monotonicity of the functions implies that the sequence of integrals \{\int_X f_n \, d\mu\} is non-decreasing, so its limit exists (finite or infinite), and \liminf_{n \to \infty} \int_X f_n \, d\mu = \lim_{n \to \infty} \int_X f_n \, d\mu.^[24] Additionally, the pointwise inequality f_n \leq f for all n, combined with the monotonicity of the Lebesgue integral for non-negative functions, gives \int_X f_n \, d\mu \leq \int_X f \, d\mu for each n. Taking the limit as n \to \infty produces

\lim_{n \to \infty} \int_X f_n \, d\mu \leq \int_X f \, d\mu.

Combining this with the inequality from Fatou's lemma results in

\int_X f \, d\mu \leq \lim_{n \to \infty} \int_X f_n \, d\mu \leq \int_X f \, d\mu,

which implies equality:

\int_X f \, d\mu = \lim_{n \to \infty} \int_X f_n \, d\mu.

This establishes the monotone convergence theorem, providing an alternative perspective to the direct construction via simple functions in Beppo Levi's original approach.^[25] The utility of this proof lies in its simplicity and generality; by leveraging Fatou's lemma, it readily extends to settings where monotonicity is relaxed, such as in the dominated convergence theorem, by incorporating an integrable dominating function to control the limsup.^[26]

Relaxing Monotonicity Assumptions

One key generalization of the monotone convergence theorem relaxes the strict monotonicity requirement by imposing a domination condition, leading to the dominated convergence theorem. Specifically, if a sequence of non-negative measurable functions f_n converges pointwise to f and there exists an integrable function g such that $0 \leq f_n \leq g for all n, then \lim_{n \to \infty} \int f_n \, d\mu = \int f \, d\mu.^[27] A further relaxation allows for sequences that are monotone almost everywhere. In this case, if f_n \uparrow f pointwise almost everywhere (i.e., f_{n+1}(x) \geq f_n(x) for all x outside a set of measure zero, and f_n \geq 0), the monotone convergence theorem still holds: \lim_{n \to \infty} \int f_n \, d\mu = \int f \, d\mu. This variant is particularly useful in L^1 spaces, where measure-zero exceptions do not affect integrability. Moreover, under the additional assumption that \sup_n \int f_n \, d\mu < \infty, the sequence converges in the L^1 norm, meaning \lim_{n \to \infty} \int |f_n - f| \, d\mu = 0.^[27] These relaxations can be established using Fatou's lemma, which provides a lower semicontinuity property for integrals without requiring full monotonicity.^[27] However, without a domination condition, even pointwise convergence of non-negative functions need not preserve integrals. A classic counterexample involves "moving bump" functions on \mathbb{R}, such as f_n(x) = \chi_{[n, n+1]}(x), which converge pointwise to 0 but satisfy \int f_n \, dx = 1 for all n, so the integrals do not converge to \int 0 \, dx = 0. Similar constructions, like f_n(x) = n \chi_{[1/n, 2/n]}(x) on [0,1], illustrate failures where no integrable dominator exists.^[27]

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