Dominated Convergence Theorem

The Dominated Convergence Theorem (DCT) is a cornerstone of modern analysis, providing one of the most powerful and widely used tools for interchanging limits and integrals. Without it, evaluating the limit of a sequence of integrals would often require ad-hoc, problem-specific methods. The theorem gives us a clear and verifiable set of conditions—centered on the existence of an integrable dominating function—that guarantees this swap is valid. Mastering the DCT is essential for advanced work in real analysis, probability theory, and mathematical physics, as it rigorously justifies techniques like "differentiating under the integral sign" that are indispensable in practice.

The Core Hypothesis: Integrable Domination

To understand the theorem, you must first grasp its hypothesis. Consider a sequence of measurable functions $f_n$ that converge pointwise to a limit function $f$. Pointwise convergence alone is insufficient to pass the limit inside the integral; you can easily construct sequences where $\lim \int f_n\ne \int \lim f_n$. The DCT strengthens the conditions by introducing a single, controlling function.

The critical requirement is that there exists an integrable dominating function $g$. This means $g$ is Lebesgue integrable (i.e., $\int |g|<\infty$), and it dominates the absolute value of every function in the sequence: $|f_n(x)|\le g(x)$ for all $n$ and almost every $x$. This condition serves two purposes. First, it prevents the sequence $f_n$ from exhibiting runaway behavior or "blowing up" in a non-integrable way. Second, it confines the sequence within an integrable envelope, allowing us to use the machinery of integration theory to control the limit.

The concept of integrability here is specific to the Lebesgue integral. A function $g$ is integrable if $\int |g|$ is finite. The dominating function $g$ acts as a universal "speed limit" for the sequence, ensuring no member of the sequence violates the integrability condition that $g$ itself satisfies.

Statement and Proof of the Theorem

With the hypothesis clear, we can state the theorem formally. Let $(X,\mathcal{F},\mu )$ be a measure space. Suppose $\{f_n\}$ is a sequence of measurable functions such that:

1. $f_n\to f$ pointwise almost everywhere.
2. There exists an integrable function $g$ such that $|f_n|\le g$ almost everywhere for all $n$.

Then, all $f_n$ and $f$ are integrable, and $$\underset{n\to \infty }{\lim }{\int }_X{f}_{n}d\mu ={\int }_{X}fd\mu .$$ Equivalently, ${\lim }_{n\to \infty }{\int }_X|f_n-f|d\mu =0$, which is convergence in $L^1$ norm.

Proof Sketch: The proof elegantly leverages Fatou's Lemma. We know $|f_n|\le g$ and $f_n\to f$ a.e., so $|f|\le g$ a.e. by properties of limits, proving integrability of $f$. To show the limit result, consider two new sequences derived from the domination: $g+f_n$ and $g-f_n$. Both are non-negative sequences because $f_n\ge -g$ and $f_n\le g$. Apply Fatou's Lemma to each: $$\int g+\int f=\int \mathrm{liminf}(g+f_n)\le \mathrm{liminf}\int (g+f_n)=\int g+\mathrm{liminf}\int f_n.$$ Canceling the finite $\int g$ gives $\int f\le \mathrm{liminf}\int f_n$. A similar argument with $g-f_n$ yields $\int g-\int f\le \int g-\mathrm{limsup}\int f_n$, which simplifies to $\mathrm{limsup}\int f_n\le \int f$. Combining these gives: $$\mathrm{limsup}\int f_n\le \int f\le \mathrm{liminf}\int f_n.$$ This forces the equality $\lim \int f_n=\int f$, completing the proof. This chain of inequalities is the central logic, showing the limit is squeezed from above and below.

Application: Evaluating Limits of Integrals

The most direct application of the DCT is computing ${\lim }_{n\to \infty }\int f_n$ by instead evaluating $\int {\lim }_{n\to \infty }{f}_n$. The challenge is always finding a suitable dominating function $g$.

Example: Consider the sequence on $[0,1]$: $f_n(x)=\frac{nx}{1+n^2{x}^2}$. It converges pointwise to $f(x)=0$. To apply the DCT, we need an integrable function $g(x)$ such that $|f_n(x)|\le g(x)$ for all $n$. Using the inequality $2ab\le a^2+b^2$ with $a=nx$ and $b=1$, we get: $$|f_n(x)|=\frac{nx}{1+n^2{x}^2}\le \frac{nx}{2nx}=\frac{1}{2},$$ for $x>0$. At $x=0$, $f_n(0)=0$. Thus, the constant function $g(x)=\frac{1}{2}$ is a valid, integrable dominator on $[0,1]$. By the DCT: $$\underset{n\to \infty }{\lim }{\int }_0^1\frac{nx}{1+n^2{x}^2}dx={\int }_0^{1}0dx=0.$$ Attempting this limit without the DCT would require a much more technical evaluation of the integral for each $n$.

Application: Differentiation Under the Integral Sign

The DCT justifies Leibniz's rule for differentiating an integral with respect to a parameter. Suppose $F(t)={\int }_{X}f(x,t)d\mu (x)$. We want to say $F^{\prime }(t)={\int }_X\frac{\partial f}{\partial t}(x,t)d\mu (x)$. The DCT provides the essential condition.

Consider the difference quotient: $\frac{F(t+h)-F(t)}{h}={\int }_X\frac{f(x,t+h)-f(x,t)}{h}d\mu (x)$. As $h\to 0$, the integrand converges pointwise to $\frac{\partial f}{\partial t}(x,t)$. If we can find an integrable function $g(x)$ that dominates the absolute value of the difference quotient for all small $h$, then the DCT allows us to swap the limit $h\to 0$ and the integral, yielding the desired result. Often, the Mean Value Theorem provides a bound: $|\frac{\partial f}{\partial t}(x,\xi )|\le g(x)$, where $g$ is integrable and independent of $h$ and $\xi$. This $g$ becomes the dominating function.

This technique is fundamental in solving integrals parametrically and in fields like thermodynamics and quantum mechanics, where integrals depend on external parameters like time or temperature.

Application in Probability Theory

In probability, the DCT is known as the Bounded Convergence Theorem when the dominator is a constant, but its general form is ubiquitous. Expectations are integrals with respect to a probability measure, so the DCT governs the interchange of limits and expectations.

Let $X_n$ be a sequence of random variables converging almost surely to $X$ (pointwise convergence a.e.). If there exists an integrable random variable $Y$ with $|X_n|\le Y$ almost surely for all $n$, then $\mathbb{E}[X_n]\to \mathbb{E}[X]$. This is vital for proving laws of large numbers and in statistical estimation.

For example, consider proving that moment generating functions are continuous. The moment generating function is $M(t)=\mathbb{E}[e^{tX}]$. To show $M(t_n)\to M(t)$ as $t_n\to t$, you view $e^{t_{n}X}$ as a sequence of random variables in $n$. Using properties of the exponential, you can often bound $|e^{t_{n}X}|\le e^{c|X|}$ for $t_n$ near $t$, and if $e^{c|X|}$ is integrable, the DCT applies. This justifies the interchange of limit and expectation, establishing continuity.

Common Pitfalls

1. Misidentifying the Dominating Function: The most frequent error is proposing a dominator $g$ that is not integrable. For example, on an infinite interval like $[1,\infty )$, the constant function $g(x)=1$ is not integrable. You must ensure $\int |g|<\infty$ over the entire measure space. Always check the integrability of your proposed $g$ on the specific domain.

1. Neglecting "Almost Everywhere" Conditions: The inequality $|f_n|\le g$ must hold for almost every $x$, not necessarily every single point. Similarly, pointwise convergence can fail on a set of measure zero. However, ignoring a problematic set of positive measure (no matter how small) invalidates the theorem. Be precise in verifying conditions hold a.e.

1. Assuming Uniform Boundedness is Enough: A sequence being uniformly bounded (i.e., $|f_n|\le M$ for all $n$) is sufficient only if the measure space is finite, like a bounded interval. On an infinite measure space, a constant bound is not an integrable function. The dominator must itself be integrable, which is a stronger condition than mere boundedness when the space has infinite measure.

1. Applying to Sequences Without Pointwise Convergence: The DCT requires a pointwise (a.e.) limit function $f$. If your sequence only converges in measure or in $L^1$, you cannot directly apply the standard DCT. There are related theorems (like the Vitali Convergence Theorem) for such cases. Ensure your sequence has a pointwise limit before attempting to use the DCT.

Summary

• The Dominated Convergence Theorem allows swapping limits and integrals when the sequence is pointwise convergent and dominated by a single, integrable function $g$.
• The proof hinges on Fatou's Lemma, using the sequences $g+f_n$ and $g-f_n$ to sandwich the limit of the integrals.
• Its primary use is to evaluate $\lim \int f_n$ by computing $\int \lim f_n$, provided a suitable integrable dominating function can be found.
• It rigorously justifies "differentiation under the integral sign," a key technique in applied mathematics and physics.
• In probability theory, it ensures the interchange of limits and expectations, underpinning continuity properties of transforms and convergence theorems for random variables.
• Always verify that your dominating function is truly integrable over the entire domain and that bounds hold almost everywhere.