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:

\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:

\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:

\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:

\begin{align} \quad \int_E f \leq \liminf_{n \to \infty} \int_E f_n \quad (**) \end{align}

- Combining $(*)$ and $(**)$ yields:

\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:

\begin{align} \quad \lim_{n \to \infty} \int_E f_n = \int_E f \quad \blacksquare \end{align}