Monotone Convergence Theorem for Integrals

A central frustration in classical calculus is the inability to freely interchange limits and integrals; you cannot always say the limit of the integrals equals the integral of the limit. The Lebesgue integral, with its more flexible foundation, provides powerful theorems that grant this permission under specific, verifiable conditions. The Monotone Convergence Theorem (MCT) is the first and perhaps most intuitive of these great convergence theorems. It states that for a sequence of functions increasing pointwise to a limit, the integral of the limit is the limit of the integrals. This result is not just a technical tool; it is the cornerstone for constructing the Lebesgue integral itself, for handling infinite series of functions, and for foundational probability theory.

The Theorem and Its Mechanics

Formally, let $(X,\mathcal{F},\mu )$ be a measure space. The Monotone Convergence Theorem states: Suppose $\{f_n\}$ is a sequence of measurable, nonnegative functions $f_n:X\to [0,\infty ]$ such that $f_1(x)\le f_2(x)\le ...$ for all $x\in X$ (this is the monotone increasing property). If $f_n$ converges pointwise to a function $f$ (i.e., $f_n(x)\to f(x)$ for each $x$), then $f$ is measurable, and $$\underset{n\to \infty }{\lim }{\int }_X{f}_{n}d\mu ={\int }_{X}fd\mu .$$ The integrals here are Lebesgue integrals, and the equality holds whether the integrals are finite or infinite. The core idea is that the integral, as a functional, respects monotone pointwise limits for nonnegative functions.

Why are the conditions necessary? Non-negativity ensures we avoid problematic cancellations of infinite areas, and the monotone increasing nature guarantees the sequence of integrals $\{\int f_n\}$ is itself a nondecreasing sequence of extended real numbers, which must have a limit (possibly $\infty$). The theorem affirms that this limit is precisely the integral of the limiting function $f$. The proof leverages the definition of the Lebesgue integral for nonnegative functions as the supremum of integrals of simple functions beneath it. The monotonicity allows one to build simple functions approximating $f$ from those approximating the $f_n$.

Application to Series of Functions

The MCT provides an immediate and powerful corollary for integrating series term-by-term. Consider a sequence $\{g_k\}$ of nonnegative measurable functions. We can form the partial sum sequence $f_n(x)={\sum }_{k=1}^n{g}_k(x)$. This sequence $\{f_n\}$ is clearly nonnegative and monotone increasing. If the infinite series converges pointwise to a function $S(x)={\sum }_{k=1}^{\infty }{g}_k(x)$, then the MCT applies directly. It yields: $${\int }_X\sum _{k=1}^{\infty }{g}_{k}d\mu ={\int }_{X}Sd\mu =\underset{n\to \infty }{\lim }{\int }_X{f}_{n}d\mu =\underset{n\to \infty }{\lim }\sum _{k=1}^n{\int }_X{g}_{k}d\mu =\sum _{k=1}^{\infty }{\int }_X{g}_{k}d\mu .$$ Thus, for nonnegative functions, summation and integration are always interchangeable. This is a stark contrast to the Riemann integral, which requires uniform convergence for a similar guarantee. This result is fundamental in harmonic analysis, probability, and any context where functions are represented as infinite sums.

Foundational Role in Constructing the Integral

The MCT is not merely a utility for using the integral; it is essential in its very definition. The standard construction of the Lebesgue integral for nonnegative functions proceeds in steps:

1. Define the integral for simple functions (finite linear combinations of characteristic functions of measurable sets).
2. For a general nonnegative measurable function $f$, define its integral as the supremum of integrals of all simple functions $s$ such that $0\le s\le f$.

While this definition is elegant, it is often intractable for direct computation. The MCT bridges this gap. One can explicitly construct a monotone increasing sequence of simple functions $s_n$ that converge pointwise to $f$. The theorem then guarantees that $\int fd\mu ={\lim }_{n\to \infty }\int s_{n}d\mu$, providing a constructive method to compute the integral that aligns with the supremal definition. In this sense, the MCT validates and completes the theoretical edifice of Lebesgue integration.

A Probability Theory Example: Expectation of a Limit

In probability, where the measure $\mu =P$ is a probability measure and integrals represent expectations, the MCT is indispensable. Let $\{Y_n\}$ be a sequence of nonnegative random variables increasing almost surely to a random variable $Y$ (e.g., $Y_n$ could be successive approximations of $Y$). The MCT (in its "almost everywhere" version) assures that $\mathbb{E}[Y]={\lim }_{n\to \infty }\mathbb{E}[Y_n]$.

A canonical application is proving the formula for the expectation of a nonnegative random variable $X$ in terms of its survival function. One defines $Y_n={\sum }_{k=1}^{n{2}^n}\frac{k-1}{2^n}\cdot 1_{\left\{\frac{k-1}{2^n}\le X<\frac{k}{2^n}\right\}}$. This sequence is nonnegative, monotone increasing, and converges pointwise to $X$. By the MCT, $\mathbb{E}[X]={\lim }_n\mathbb{E}[Y_n]$. Evaluating the limit of the expectations leads directly to the well-known result: $$\mathbb{E}[X]={\int }_0^{\infty }P(X>t)dt.$$ This derivation rigorously relies on the interchange of limit and integral sanctioned by the MCT.

Common Pitfalls

1. Ignoring Non-Negativity: The theorem fails if functions can take on negative values. Consider $f_n=-\frac{1}{n}\cdot 1_{[0,n]}$ on $\mathbb{R}$ with Lebesgue measure. This sequence is monotone increasing (from $-1$ toward $0$), converges pointwise to $0$, but $\int f_n=-1$ for all $n$, while $\int 0=0$. The limit of the integrals ($-1$) does not equal the integral of the limit ($0$). The non-negativity condition is crucial.

1. Confusing Types of Convergence: The theorem requires pointwise (or almost everywhere) monotone convergence. It does not hold under only measure convergence or $L^1$-convergence. You can have sequences converging in measure that are monotone, yet the conclusion of the MCT fails, highlighting the strength of the pointwise hypothesis.

1. Overlooking the "Monotone Increasing" Condition: The sequence must be nondecreasing for every $x$. If the sequence is not monotone, you must use other tools like Fatou's Lemma or the Dominated Convergence Theorem. Applying the MCT to a non-monotone sequence, even if it consists of nonnegative functions, will lead to an incorrect conclusion.

1. Misapplication to Decreasing Sequences: The theorem is not symmetric. For a monotone decreasing sequence of nonnegative functions, a similar conclusion holds, but it requires an additional integrability condition (i.e., that $\int f_1<\infty$) to avoid issues with $\infty -\infty$ ambiguities. Directly applying the standard MCT to a decreasing sequence is a common error.

Summary

• The Monotone Convergence Theorem permits the interchange of limit and integral for sequences of nonnegative, measurable functions that increase pointwise to a limit function.
• It is a foundational pillar used to construct the Lebesgue integral itself, providing a bridge from integrals of simple functions to integrals of general nonnegative functions.
• A direct corollary is that infinite series of nonnegative functions can always be integrated term-by-term, a result with wide applications in analysis and probability.
• In probability theory, the MCT is frequently used to compute expectations of limiting random variables and to prove fundamental formulas linking expectation to the survival function.
• The theorem's conditions are strict: non-negativity, pointwise monotone increase, and measurability. Violating any of these can lead to false results, making careful verification essential before application.