The Lebesgue Dominated Convergence Theorem

The Lebesgue Dominated Convergence Theorem

 Theorem (The Lebesgue Dominated Convergence Theorem): Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of Lebesgue measurable functions defined on a Lebesgue measurable set $E$. Suppose that: 1) There exists a nonnegative Lebesgue integrable function $g$ on $E$ such that $|f_n(x)| \leq g(x)$ for all $x \in E$. 2) $(f_n(x))_{n=1}^{\infty}$ converges pointwise to $f(x)$ almost everywhere on $E$. Then $f$ is Lebesgue integrable on $E$ and $\displaystyle{\lim_{n \to \infty} \int_E f_n = \int_E f}$.
• Proof: From (1), since $g$ is a Lebesgue integrable function on $E$ such that $|f_n(x)| \leq g(x)$ for all $x \in E$ we have from The Comparison Test for Lebesgue Integrability that for each $n \in \mathbb{N}$, the Lebesgue measurable function $f_n$ is Lebesgue integrable on $E$.
• Furthermore, we also have that $|f(x)| \leq g(x)$ for all $x \in E$. Since $f$ is Lebesgue measurable (as it is a pointwise limit of Lebesgue measurable functions) we also have by the comparison test that $f$ is Lebesgue integrable on $E$.
• Now consider the sequence of functions $(g(x) - f_n(x))_{n=1}^{\infty}$. This is a sequence of nonnegative Lebesgue measurable functions defined on a Lebesgue measurable set $E$ that converges pointwise to $g(x) - f(x)$. So by Fatou's Lemma for Nonnegative Lebesgue Measurable Functions we have that:
(1)
\begin{align} \quad \int_E (g - f) \leq \liminf_{n \to \infty} \int_E (g - f_n) \end{align}
• By the linearity of the Lebesgue integral for nonnegative Lebesgue measurable functions we have that:
(2)
\begin{align} \quad \int_E g - \int_E f & \leq \liminf_{n \to \infty} \int_E (g - f_n) \\ & \leq \int_E g - \limsup_{n \to \infty} \int_E f_n \end{align}
• Therefore:
(3)
\begin{align} \quad -\int_E f \leq - \limsup_{n \to \infty} \int_E f_n \quad \Leftrightarrow \quad \int_E f \geq \limsup_{n \to \infty} \int_E f_n \quad (*) \end{align}
• Now consider the sequence of functions $(g(x) + f_n(x))_{n=1}^{\infty}$. Since $|f_n(x)| \leq g(x)$ for all $x \in E$ we have that $|f_n(x)| + f_n(x) \leq g(x) + f_n(x)$. So $(g(x) + f_n(x))_{n=1}^{\infty}$ is a nonnegative sequence of Lebesgue measurable functions that converges pointwise to $g(x) + f(x)$. So by Fatou's lemma we have that:
(4)
\begin{align} \quad \int_E (g + f) \leq \liminf_{n \to \infty} \int_E (g + f_n) \\ \end{align}
• By the linearity of the Lebesgue integral for nonnegative Lebesgue measurable functions we have that:
(5)
\begin{align} \quad \int_E g + \int_E f \leq \int_E g + \liminf_{n \to \infty} \int_E f_n \quad \Leftrightarrow \quad \int_E f \leq \liminf_{n \to \infty} \int_E f_n \quad (**) \end{align}
• Combining $(*)$ and $(**)$ yields:
(6)
\begin{align} \quad \limsup_{n \to \infty} \int_E f_n \leq \int_E f \leq \liminf_{n \to \infty} \int_E f \end{align}
• By definition the limit inferior is always less than or equal to the limit superior of a sequence of numbers. Therefore the inequality above implies that:
(7)
\begin{align} \quad \lim_{n \to \infty} \int_E f_n = \int_E f \quad \blacksquare \end{align}