The Generalized Lebesgue Dominated Convergence Theorem

# The Generalized Lebesgue Dominated Convergence Theorem

 Theorem (The Generalized Lebesgue Dominated Convergence Theorem): Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of Lebesgue measurable functions defined on a Lebesgue measurable set $E$, and let $(g_n(x))_{n=1}^{\infty}$ be a sequence of nonnegative Lebesgue measurable functions defined on $E$. Suppose that: 1) $|f_n(x)| \leq |g_n(x)|$ for all $n \in \mathbb{N}$ and for all $x \in E$. 2) $(f_n(x))_{n=1}^{\infty}$ converges pointwise almost everywhere to $f(x)$ and $(g_n(x))_{n=1}^{\infty}$ converges pointwise almost everywhere to $g(x)$. 3) $\displaystyle{\lim_{n \to \infty} \int_E g_n = \int_E g < \infty}$. Then $f$ is Lebesgue integrable on $E$ and $\displaystyle{\lim_{n \to \infty} \int_E f_n = \int_E f}$.
• Proof: Since each $g_n$ is a nonnegative Lebesgue measurable function we see that each $g_n$ is Lebesgue integrable. So by The Comparison Test for Lebesgue Integrability we have that each $f_n$ is Lebesgue integrable on $E$. Furthermore, $g$ and $f$ are also Lebesgue integrable on $E$.
• Consider the sequence $(g_n(x) - f_n(x))_{n=1}^{\infty}$ of Lebesgue integrable functions. This sequence converges pointwise almost everywhere to $g(x) - f(x)$. Since $|f_n(x)| \leq g_n(x)$ for all $x \in E$ we have that $|f_n(x)| - f_n(x) | \leq g_n(x) - f_n(x)$ for all $x \in E$ so this is a sequence of nonnegative Lebesgue integrable functions on $E$. By Fatou's Lemma for Nonnegative Lebesgue Measurable Functions and linearity for the Lebesgue integral of nonnegative Lebesgue measurable functions we have that:
(1)
\begin{align} \quad \int_E g - \int_E f = \int_E (g - f) \leq \liminf_{n \to \infty} \int_E (g_n - f_n) = \lim_{n \to \infty} \int_E g_n - \limsup_{n \to \infty} \int_E f_n = \int_E g - \limsup_{n \to \infty} \int_E f_n \end{align}
• Therefore:
(2)
\begin{align} \quad \int_E f \geq \limsup_{n \to \infty} \int_E f_n \quad (*) \end{align}
• Now consider the sequence $(g_n(x)) + f_n(x))_{n=1}^{\infty}$ of Lebesgue integrable functions. This sequence converges pointwise almost everywhere to $g(x) + f(x)$. Since $|f_n(x)| \leq g_n(x)$ for all $x \in E$ we have that $|f_n(x)| + f_n(x) \leq g_n(x) + f_n(x)$ for all $x \in E$ so this is a sequence of nonnegative Lebesgue integrable functions on $E$. By Fatou's Lemma and linearity for the Lebesgue integrable of nonnegative Lebesgue measurable functions we have that:
(3)
\begin{align} \quad \int_E g + \int_E f = \int_E (g + f) \leq \liminf_{n \to \infty} \int_E (g_n + f_n) = \lim_{n \to \infty} \int_E g_n + \liminf_{n \to \infty} \int_E f_n = \int_E g + \liminf_{n \to \infty} \int_E f_n \end{align}
• Therefore:
(4)
\begin{align} \quad \int_E f \leq \liminf_{n \to \infty} \int_E f_n \quad (**) \end{align}
• Combining $(*)$ and $(**)$ yields:
(5)
\begin{align} \quad \limsup_{n \to \infty} \int_E f \leq \int_E f \leq \liminf_{n \to \infty} \int_E f \end{align}
• The above inequality implies that:
(6)
\begin{align} \quad \lim_{n \to \infty} \int_E f_n = \int_E f \quad \blacksquare \end{align}